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{{about|the field of science}}
{{About|the binary relation|the graph vertex ordering|Depth-first search|other uses}}
{{pp-semi-indef}}{{pp-move-indef}}
{{Use dmy dates|date=January 2012}}
[[File:CollageFisica.jpg|300px|thumb|Various examples of physical phenomena. {{see|Outline of physics}}]]


'''Physics''' (from {{lang-grc|φυσική (ἐπιστήμη)|phusikḗ (epistḗmē)|knowledge of nature}}, from {{lang|grc|φύσις}} ''phúsis'' "nature"<!--
In [[mathematics]], especially in [[order theory]], a '''preorder''' or '''quasi-order''' is a [[binary relation]] that is [[reflexive relation|reflexive]] and [[transitive relation|transitive]]. All [[partial order]]s and [[equivalence relation]]s are preorders, but preorders are more general.
  --><ref name="etymonline-physics">{{cite web |title=physics |url=http://www.etymonline.com/index.php?term=physics&allowed_in_frame=0 |publisher=[[Online Etymology Dictionary]]}}</ref><ref name="etymonline-physic">{{cite web |title=physic |url=http://www.etymonline.com/index.php?term=physic&allowed_in_frame=0 |publisher=[[Online Etymology Dictionary]]}}</ref><ref name="LSJ">{{LSJ|fu/sis|φύσις}}, {{LSJ|fusiko/s|φυσική}}, {{LSJ|e)pisth/mh|ἐπιστήμη|ref}}</ref><!--
          -->) is the [[natural science]] that involves the study of [[matter]]<!--
  --><ref name="feynmanleightonsands1963-atomic">At the start of ''[[The Feynman Lectures on Physics]]'', [[Richard Feynman]] offers the [[Atomic theory|atomic hypothesis]] as the single most prolific scientific concept: "If, in some cataclysm, all [] scientific knowledge were to be destroyed [save] one sentence&nbsp;[...] what statement would contain the most information in the fewest words? I believe it is&nbsp;[...] that ''all things are made up of atoms&nbsp;– little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another''&nbsp;..." {{harv|Feynman|Leighton|Sands|1963|p=I-2}}</ref> <!--
          -->and its [[Motion (physics)|motion]] through [[Spacetime|space and time]], along with related concepts such as [[energy]] and [[force]].<!--
  --><ref name="maxwell1878-physicalscience">"Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." {{harv|Maxwell|1878|p=9}}</ref> <!--
          -->More broadly, it is the general analysis of [[nature]], conducted in order to understand how the [[universe]] behaves.<!--
  -->{{efn|The term 'universe' is defined as everything that physically exists: the entirety of space and time, all forms of matter, energy and momentum, and the physical laws and constants that govern them. However, the term 'universe' may also be used in slightly different contextual senses, denoting concepts such as the [[cosmos]] or the [[World (philosophy)|philosophical world]].}}<ref name="youngfreedman2014p9">{{harvnb|Young|Freedman|2014|p=9}}</ref><ref name="holzner2003-physics">"Physics is the study of your world and the world and universe around you." {{harv|Holzner|2006|p=7}}</ref>


Physics is one of the oldest [[academic discipline]]s, perhaps the oldest through its inclusion of [[astronomy]].<ref name="krupp2003">{{harvnb|Krupp|2003}}</ref> Over the last two millennia, physics was a part of [[natural philosophy]] along with [[chemistry]], certain branches of [[mathematics]], and [[biology]], but during the [[Scientific Revolution]] in the 17th century, the [[natural science]]s emerged as unique [[research]] programs in their own right.{{efn|[[Francis Bacon]]'s 1620 ''[[Novum Organum]]'' was critical in the [[History of scientific method|development of scientific method]].}} Physics intersects with many [[interdisciplinary]] areas of research, such as [[biophysics]] and [[quantum chemistry]], and the boundaries of physics are not [[demarcation problem|rigidly defined]]. New ideas in physics often explain the fundamental mechanisms of other sciences<ref name="youngfreedman2014p9" /> while opening new avenues of research in areas such as [[mathematics]] and [[philosophy]].
The name 'preorder' comes from the idea that preorders are 'almost' (partial) orders, but not quite; they're neither [[anti-symmetric relation|anti-symmetric]] nor [[symmetric relation|symmetric]]. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not anti-symmetric, some of the ordinary intuition that a student may have with regards to the symbol ≤ may not apply. On the other hand, a pre-order can be used, in a straightforward fashion, to define a partial order and an equivalence relation.  Doing so, however, is not always useful or worth-while, depending on the problem domain being studied.


Physics also makes significant contributions through advances in new [[technology|technologies]] that arise from theoretical breakthroughs. For example, advances in the understanding of [[electromagnetism]] or [[nuclear physics]] led directly to the development of new products that have dramatically transformed modern-day [[society]], such as [[television]], [[computer]]s, [[domestic appliance]]s, and [[nuclear weapon]]s;<ref name="youngfreedman2014p9" /> advances in [[thermodynamics]] led to the development of [[industrialization]], and advances in [[mechanics]] inspired the development of [[calculus]].
In words, when ''a'' ≤ ''b'', one may say that ''b'' ''covers'' ''a'' or that ''b'' ''precedes'' ''a'', or that ''b'' ''reduces'' to ''a''. Occasionally, the notation ← or <math>\lesssim</math> is used instead of ≤.


==History==
To every preorder, there corresponds a [[directed graph]], with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs may be [[cyclic graph]]s: preorders may have cycles in them.  A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a [[directed acyclic graph]].  A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.  In general, a preorder may have many disconnected components.  The [[diamond lemma]] is an important result for certain kinds of preorders.
{{Main|History of physics}}


=== Ancient astronomy ===
Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.{{Citation needed|date=July 2010}}
{{Main|History of astronomy}}
[[File:Senenmut-Grab.JPG|thumb|right|Ancient [[Egyptian astronomy]] is evident in monuments like the [[Astronomical ceiling of Senemut Tomb|ceiling of Senemut's tomb]] from the [[Eighteenth Dynasty of Egypt]].]]
[[Astronomy]] is the oldest of the [[natural science]]s. The earliest civilizations dating back to beyond 3000&nbsp;BCE, such as the [[Sumer]]ians, [[Ancient Egyptians]], and the [[Indus Valley Civilization]], all had a predictive knowledge and a basic understanding of the motions of the [[Sun]], [[Moon]], and [[star]]s. The stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy.<ref name="krupp2003"/>


According to [[Asger Aaboe]], the origins of [[Western world|Western]] [[astronomy]] can be found in [[Mesopotamia]], and all Western efforts in the [[exact sciences]] are descended from late [[Babylonian astronomy]].<ref name ="aaboe1991">{{harvnb|Aaboe|1991}}</ref> [[Egyptian astronomy|Egyptian astronomers]] left monuments showing knowledge of the constellations and the motions of the celestial bodies,<ref name="clagett1995">{{harvnb|Clagett|1995}}</ref> while [[Ancient Greek poetry|Greek poet]] [[Homer]] wrote of various celestial objects in his ''[[Iliad]]'' and ''[[Odyssey]]''; later [[Greek astronomy|Greek astronomers]] provided names, which are still used today, for most constellations visible from the [[northern hemisphere]].<ref name="thurston1994">{{harvnb|Thurston|1994}}</ref>
==Formal definition==


=== Natural philosophy ===
Consider some [[Set (mathematics)|set]] ''P'' and a [[binary relation]] ≤ on ''P''. Then ≤ is a '''preorder''', or '''quasiorder''', if it is [[reflexive relation|reflexive]] and [[transitive relation|transitive]], i.e., for all ''a'', ''b'' and ''c'' in ''P'', we have that:
{{main|Natural philosophy}}
[[Natural philosophy]] has its origins in [[Greece]] during the [[Archaic Greece|Archaic period]], (650 BC – 480 BC), when [[Presocratics|Pre-Socratic philosophers]] like [[Thales]] rejected [[Methodological naturalism|non-naturalistic]] explanations for natural phenomena and proclaimed that every event had a natural cause.<ref name="singer2008p35">{{harvnb|Singer|2008|p=35}}</ref> They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment;<ref name="lloyd1970pp108-109">{{harvnb|Lloyd|1970|pp=108–109}}</ref> for example, [[atomism]] was found to be correct approximately 2000 years after it was first proposed by [[Leucippus]] and his pupil [[Democritus]].<ref name="about-atomism">{{cite web
|last=Gill
|first=N.S.
|title=Atomism - Pre-Socratic Philosophy of Atomism
|work=Ancient/Classical History
|publisher=[[About.com]]
|url=http://ancienthistory.about.com/od/presocraticphiloso/p/Atomism.htm
|accessdate=2014-04-01
|ref=harv}}</ref>


=== Classical physics ===
:''a'' ≤ ''a'' (reflexivity)
{{main|Classical physics}}
: if ''a'' ≤ ''b'' and ''b'' ≤ ''c'' then ''a'' ≤ ''c'' (transitivity)
[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|right|upright|[[Sir Isaac Newton]] (1643–1727), whose [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|universal gravitation]] were major milestones in classical physics]]
Physics became a separate science when [[early modern Europe]]ans used experimental and quantitative methods to discover what are now considered to be the [[laws of physics]].<ref name="benchaim2004">{{harvnb|Ben-Chaim|2004}}</ref>


Major developments in this period include the replacement of the [[geocentric model]] of the solar system with the helio-centric [[Copernican model]], the [[Kepler's laws|laws governing the motion of planetary bodies]] determined by [[Johannes Kepler]] between 1609 and 1619, pioneering work on [[telescope]]s and [[observational astronomy]] by [[Galileo Galilei]] in the 16th and 17th Centuries, and [[Isaac Newton]]'s discovery and unification of the [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|universal gravitation]] that would come to bear his name.<ref>{{harvnb|Guicciardini|1999}}</ref> Newton also developed [[calculus]],{{efn|Calculus was independently developed at around the same time by [[Gottfried Wilhelm Leibniz]]; while Leibniz was the first to publish his work, and developed much of the notation used for calculus today, Newton was the first to develop calculus and apply it to physical problems. See also [[Leibniz–Newton calculus controversy]]}} the mathematical study of change, which provided new mathematical methods for solving physical problems.<ref name="allen1997">{{harvnb|Allen|1997}}</ref>
''Note that an alternate definition of preorder requires the relation to be [[irreflexive relation|irreflexive]]. However, as this article is examining preorders as a logical extension of non-strict partial orders, the current definition is more intuitive.''


The discovery of new laws in [[thermodynamics]], [[chemistry]], and [[electromagnetics]] resulted from greater research efforts during the [[Industrial Revolution]] as energy needs increased.<ref name="schoolscience-industrialrevolution">{{cite web
A set that is equipped with a preorder is called a '''preordered set''' (or '''proset''').
|title=The Industrial Revolution
|publisher=Schoolscience.org, [[Institute of Physics]]
|url=http://resources.schoolscience.co.uk/IoP/14-16/biogs/biogs5.html


|accessdate=2014-04-01
If a preorder is also [[antisymmetric relation|antisymmetric]], that is, ''a'' ≤ ''b'' and ''b'' ≤ ''a'' implies ''a'' = ''b'', then it is a [[partially ordered set|partial order]].
|ref=harv}}</ref> The laws comprising classical physics remain very widely used for objects on everyday scales travelling at non-relativistic speeds, since they provide a very close approximation in such situations, and theories such as [[quantum mechanics]] and the [[theory of relativity]] simplify to their classical equivalents at such scales. However, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century.


=== Modern physics ===
On the other hand, if it is [[symmetric relation|symmetric]], that is, if ''a'' ≤ ''b'' implies ''b'' ≤ ''a'', then it is an [[equivalence relation]].
{{main|Modern physics}}
{{see also|History of special relativity|History of quantum mechanics}}
[[File:Einstein1921 by F Schmutzer 2.jpg|thumb|right|upright|[[Albert Einstein]] (1879–1955), whose work on the [[photoelectric effect]] and the [[theory of relativity]] led to a revolution in 20th century physics]]
[[File:Max Planck (Nobel 1918).jpg|thumb|left|upright|[[Max Planck]] (1858–1947), the originator of the theory of [[quantum mechanics]]]]
[[Modern physics]] began in the early 20th century with the work of [[Max Planck]] in [[Quantum mechanics|quantum theory]] and [[Albert Einstein]]'s [[theory of relativity]]. Both of these theories came about due to inaccuracies in classical mechanics in certain situations. [[Classical mechanics]] predicted a varying [[speed of light]], which could not be resolved with the constant speed predicted by [[Maxwell's equations]] of electromagnetism; this discrepancy was corrected by Einstein's theory of [[special relativity]], which replaced classical mechanics for fast-moving bodies and allowed for a constant speed of light.<ref name="oconnorrobertson1996-relativity">{{harvnb|O'Connor|Robertson|1996a}}</ref> [[Black body radiation]] provided another problem for classical physics, which was corrected when Planck proposed that light comes in individual packets known as [[photons]]; this, along with the [[photoelectric effect]] and a complete theory predicting discrete [[energy levels]] of [[electron orbitals]], led to the theory of quantum mechanics taking over from classical physics at very small scales.<ref name="oconnorrobertson1996-quantum">{{harvnb|O'Connor|Robertson|1996b}}</ref>


[[Quantum mechanics]] would come to be pioneered by [[Werner Heisenberg]], [[Erwin Schrödinger]] and [[Paul Dirac]].<ref name="oconnorrobertson1996-quantum"/> From this early work, and work in related fields, the [[Standard Model of particle physics]] was derived.<ref name="donut2001">{{harvnb|DONUT|2001}}</ref> Following the discovery of a particle with properties consistent with the [[Higgs boson]] at [[CERN]] in 2012,<ref name="cho2012">{{harvnb|Cho|2012}}</ref> all [[fundamental particles]] predicted by the standard model, and no others, appear to exist; however, [[physics beyond the Standard Model]], with theories such as [[supersymmetry]], is an active area of research.{{citation needed|date=August 2014}}
A preorder which is preserved in all contexts (i.e. respected by all functions on ''P'') is called a '''precongruence'''.
A precongruence which is also [[symmetric relation|symmetric]] (i.e. is an [[equivalence relation]]) is a [[congruence relation]].


==Philosophy==
Equivalently, a preordered set ''P'' can be defined as a [[category theory|category]] with [[object (category theory)|objects]] the elements of ''P'', and each [[hom-set]] having at most one element (one for objects which are related, zero otherwise).
{{Main|Philosophy of physics}}
In many ways, physics stems from [[ancient Greek philosophy]]. From [[Thales]]' first attempt to characterize matter, to [[Democritus]]' deduction that matter ought to reduce to an invariant state, the [[Ptolemaic astronomy]] of a crystalline [[firmament]], and Aristotle's book ''[[Physics (Aristotle)|Physics]]'' (an early book on physics, which attempted to analyze and define motion from a philosophical point of view), various Greek philosophers advanced their own theories of nature. Physics was known as [[natural philosophy]] until the late 18th century.{{citation needed|date=August 2014}}


By the 19th century, physics was realized as a discipline distinct from philosophy and the other sciences. Physics, as with the rest of science, relies on [[philosophy of science]] to give an adequate description of the scientific method.<ref name="rosenberg2006ch1">{{harvnb|Rosenberg|2006|loc=Chapter 1}}</ref> The scientific method employs ''[[A priori and a posteriori|a priori reasoning]]'' as well as ''[[Empirical evidence|a posteriori]]'' reasoning and the use of [[Bayesian inference]] to measure the validity of a given theory.<ref name="godfreysmith2003ch14">{{harvnb|Godfrey-Smith|2003|loc=Chapter 14: "Bayesianism and Modern Theories of Evidence"}}</ref>
Alternately, a preordered set can be understood as an [[enriched category]], enriched over the category '''2''' = (0→1).


The development of physics has answered many questions of early philosophers, but has also raised new questions. Study of the philosophical issues surrounding physics, the philosophy of physics, involves issues such as the nature of [[space]] and [[time]], [[determinism]], and metaphysical outlooks such as [[empiricism]], [[naturalism (philosophy)|naturalism]] and [[Philosophical realism|realism]].<ref name="godfreysmith2003ch15">{{harvnb|Godfrey-Smith|2003|loc=Chapter 15: "Empiricism, Naturalism, and Scientific Realism?"}}</ref>
==Examples==
* The [[reachability]] relationship in any [[directed graph]] (possibly containing cycles) gives rise to a preorder, where ''x''&nbsp;≤&nbsp;''y'' in the preorder if and only if there is a path from ''x'' to ''y'' in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from ''x'' to ''y'' for every pair (''x'', ''y'') with ''x''&nbsp;≤&nbsp;''y''). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of [[directed acyclic graph]]s, directed graphs with no cycles, gives rise to [[partially ordered set]]s (preorders satisfying an additional anti-symmetry property).
* Every [[finite topological space]] gives rise to a preorder on its points, in which ''x'' ≤ ''y'' if and only if ''x'' belongs to every neighborhood of ''y'', and every finite preorder can be formed as the [[Specialization_(pre)order|specialization preorder]] of a topological space in this way. That is, there is a [[bijection|1-to-1 correspondence]] between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not 1-to-1.
* A [[net (mathematics)|net]] is a [[directed set|directed]] preorder, that is, each pair of elements has an [[upper bound]].  The definition of convergence via nets is important in [[topology]], where preorders cannot be replaced by [[partially ordered set]]s without losing important features.
* The relation defined by <math>x \le y</math> [[iff]] <math>f(x) \le f(y)</math>, where ''f'' is a function into some preorder.
* The relation defined by <math>x \le y</math> [[iff]] there exists some [[injective function|injection]] from ''x'' to ''y''. Injection may be replaced by [[surjection]], or any type of structure-preserving function, such as [[ring homomorphism]], or [[permutation]].
* The [[embedding]] relation for countable [[total order]]ings.
* The [[graph-minor]] relation in [[graph theory]].
* A [[category (mathematics)|category]] with at most one [[morphism]] between any pair of objects is a preorder. Such categories are called [[thin category|thin]]. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.


Many physicists have written about the philosophical implications of their work, for instance [[Laplace]], who championed [[causal determinism]],<ref name="laplace1951">{{harvnb|Laplace|1951}}</ref> and [[Erwin Schrödinger]], who wrote on [[quantum mechanics]].<ref name="schroedinger1983">{{harvnb|Schrödinger|1983}}</ref><ref name="schroedinger1995">{{harvnb|Schrödinger|1995}}</ref> The mathematical physicist [[Roger Penrose]] has been called a [[Platonism|Platonist]] by [[Stephen Hawking]],<ref name="hawkingpenrose1996p4">"I think that Roger is a Platonist at heart but he must answer for himself." {{harv|Hawking|Penrose|1996|p=4}}</ref> a view Penrose discusses in his book, ''[[The Road to Reality]]''.<ref name="penrose2004">{{harvnb|Penrose|2004}}</ref> Hawking refers to himself as an "unashamed reductionist" and takes issue with Penrose's views.<ref name="penroseshimonycartwrighthawking1997">{{harvnb|Penrose|Shimony|Cartwright|Hawking|1997}}</ref>
In computer science, one can find examples of the following preorders.
* The [[Subtype|subtyping]] relations are usually preorders.
* [[Simulation preorder]]s are preorders (hence the name).
* [[Reduction relation]]s in [[abstract rewriting system]]s.


==Core theories==
Example of a [[Strict weak ordering#Total preorders|total preorder]]:
{{Further2|[[Branches of physics]], [[Outline of physics]]}}
* [[Preference]], according to common models.
Though physics deals with a wide variety of systems, certain theories are used by all physicists. Each of these theories were experimentally tested numerous times and found correct as an approximation of nature (within a certain domain of validity). For instance, the theory of [[Classical physics|classical]] mechanics accurately describes the motion of objects, provided they are much larger than [[atom]]s and moving at much less than the [[speed of light]]. These theories continue to be areas of active research, and a remarkable aspect of classical mechanics known as [[chaos theory|chaos]] was discovered in the 20th century, three centuries after the original formulation of classical mechanics by [[Isaac Newton]] (1642–1727).


These central theories are important tools for research into more specialised topics, and any physicist, regardless of their specialisation, is expected to be literate in them. These include [[classical mechanics]], [[quantum mechanics]], [[thermodynamics]] and [[statistical mechanics]], [[electromagnetism]], and [[special relativity]].
==Uses==
Preorders play a pivotal role in several situations:
* Every preorder can be given a topology, the [[Alexandrov topology]]; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
* Preorders may be used to define [[interior algebra]]s.
* Preorders provide the [[Kripke semantics]] for certain types of [[modal logic]].


===Classical physics===
==Constructions==
{{Main|Classical physics}}
[[File:Prediction of sound scattering from Schroeder Diffuser.jpg|thumb|Classical physics implemented in an [[acoustic engineering]] model of sound reflecting from an acoustic diffuser]]


[[Classical physics]] includes the traditional branches and topics that were recognised and well-developed before the beginning of the 20th century—[[classical mechanics]], [[acoustics]], [[optics]], [[thermodynamics]], and [[electromagnetism]]. [[Classical mechanics]] is concerned with bodies acted on by [[force]]s and bodies in [[motion (physics)|motion]] and may be divided into [[statics]] (study of the forces on a body or bodies not subject to an acceleration), [[kinematics]] (study of motion without regard to its causes), and [[Analytical dynamics|dynamics]] (study of motion and the forces that affect it); mechanics may also be divided into [[solid mechanics]] and [[fluid mechanics]] (known together as [[continuum mechanics]]), the latter including such branches as [[Fluid statics|hydrostatics]], [[Fluid dynamics|hydrodynamics]], [[aerodynamics]], and [[pneumatics]]. [[Acoustics]] is the study of how [[sound]] is produced, controlled, transmitted and received.<ref name="britannica-acoustics">{{cite encyclopedia|title=acoustics|url=http://www.britannica.com/EBchecked/topic/4044/acoustics|encyclopedia=Encyclopædia Britannica|accessdate=14 June 2013|ref=harv}}</ref>  Important modern branches of acoustics include [[ultrasonics]], the study of sound waves of very high frequency beyond the range of human hearing; [[bioacoustics]] the physics of animal calls and hearing,<ref>{{cite web |url=http://www.bioacoustics.info/ |title=Bioacoustics – the International Journal of Animal Sound and its Recording |publisher=Taylor & Francis |accessdate=31 July 2012}}</ref> and [[electroacoustics]], the manipulation of audible sound waves using electronics.<ref>{{cite web|last=Acoustical Society of America|title=Acoustics and You (A Career in Acoustics?)|url=http://asaweb.devcloud.acquia-sites.com/education_outreach/careers_in_acoustics|accessdate=21 May 2013}}</ref> [[Optics]], the study of [[light]], is concerned not only with [[visible light]] but also with [[infrared]] and [[ultraviolet radiation]], which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light. [[Heat]] is a form of [[energy]], the internal energy possessed by the particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy. [[Electricity]] and [[magnetism]] have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an [[electric current]] gives rise to a [[magnetic field]], and a changing magnetic field induces an electric current. [[Electrostatics]] deals with [[electric charge]]s at rest, [[Classical electromagnetism|electrodynamics]] with moving charges, and [[magnetostatics]] with magnetic poles at rest.
Every binary relation R on a set S can be extended to a preorder on S by taking the [[transitive closure]] and [[Binary relation#Operations on binary relations|reflexive closure]], R<sup>+=</sup>. The transitive closure indicates path connection in R: ''x'' R<sup>+</sup> ''y'' if and only if there is an R-[[Path (graph theory)|path]] from ''x'' to y.


===Modern physics===
Given a preorder <math>\lesssim</math> on S one may define an [[equivalence relation]] ~ on S such that ''a'' ~ ''b'' if and only if ''a'' <math>\lesssim</math> ''b'' and ''b'' <math>\lesssim</math> ''a''.  (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.)
{{Main|Modern physics}}
{{Modern Physics}}
[[File:Solvay conference 1927.jpg|thumb|left|[[Solvay Conference]] of 1927, with prominent physicists such as [[Albert Einstein]], [[Werner Heisenberg]], [[Max Planck]], [[Hendrik Lorentz]], [[Niels Bohr]], [[Marie Curie]], [[Erwin Schrödinger]] and [[Paul Dirac]].]]


Classical physics is generally concerned with matter and energy on the normal scale of observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on a very large or very small scale. For example, [[Atomic physics|atomic]] and [[nuclear physics]] studies matter on the smallest scale at which [[chemical element]]s can be identified. The [[Particle physics|physics of elementary particles]] is on an even smaller scale since it is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in large [[particle accelerator]]s. On this scale, ordinary, commonsense notions of space, time, matter, and energy are no longer valid.
Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all [[equivalence class]]es of ~.  Note that if the preorder is R<sup>+=</sup>, S / ~ is the set of R-[[Cycle (graph theory)|cycle]] equivalence classes: ''x'' ∈ [''y''] if and only if ''x'' = ''y'' or ''x'' is in an R-cycle with y.  In any case, on S / ~ we can define [''x''] ≤ [''y''] if and only if ''x'' <math>\lesssim</math> ''y''. By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.


The two chief theories of modern physics present a different picture of the concepts of space, time, and matter from that presented by classical physics. [[Quantum mechanics|Quantum theory]] is concerned with the discrete, rather than continuous, nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The [[theory of relativity]] is concerned with the description of phenomena that take place in a [[frame of reference]] that is in motion with respect to an observer; the [[special relativity|special theory of relativity]] is concerned with relative uniform motion in a straight line and the [[General relativity|general theory of relativity]] with accelerated motion and its connection with [[gravitation]]. Both quantum theory and the theory of relativity find applications in all areas of modern physics.
Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).


===Difference between classical and modern physics===
For a preorder "<math>\lesssim</math>", a relation "<" can be defined as ''a'' < ''b'' if and only if (''a'' <math>\lesssim</math> ''b'' and not ''b'' <math>\lesssim</math> ''a''), or equivalently, using the equivalence relation introduced above, (''a'' <math>\lesssim</math> ''b'' and not ''a'' ~ ''b''). It is a [[strict partial order]]; every strict partial order can be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as ''a'' < ''b'' if and only if (''a'' ≤ ''b'' and ''a'' ≠ ''b'').
[[File:Modernphysicsfields.svg|thumb|350px|left|The basic domains of physics]]


While physics aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking, the laws of [[classical physics]] accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match their predictions. [[Albert Einstein]] contributed the framework of [[special relativity]], which replaced notions of [[absolute time and space]] with [[spacetime]] and allowed an accurate description of systems whose components have speeds approaching the speed of light. [[Max Planck]], [[Erwin Schrödinger]], and others introduced [[quantum mechanics]], a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales. Later, [[quantum field theory]] unified [[quantum mechanics]] and [[special relativity]]. [[General relativity]] allowed for a dynamical, curved [[spacetime]], with which highly massive systems and the large-scale structure of the universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of [[quantum gravity]] are being developed.
(Alternatively, for a preorder "<math>\lesssim</math>", a relation "<" can be defined as ''a'' < ''b'' if and only if (''a'' <math>\lesssim</math> ''b'' and ''a'' ≠ ''b''). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before.)
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==Relation to other fields==
Conversely we have ''a'' <math>\lesssim</math> ''b'' if and only if ''a'' < ''b'' or ''a'' ~ ''b''. This is the reason for using the notation "<math>\lesssim</math>"; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that ''a'' ≤ ''b'' implies that ''a'' < ''b'' or ''a'' = ''b''.
[[File:Pahoeoe fountain original.jpg|thumb|This [[parabola]]-shaped [[lava flow]] illustrates the application of mathematics in physics—in this case, [[Galileo]]'s [[law of falling bodies]].]]
[[File:Physics and other sciences.png|thumb|left|Mathematics and ontology are used in physics. Physics is used in chemistry and cosmology.]]


===Prerequisites===
Note that with this construction multiple preorders "<math>\lesssim</math>" can give the same relation "<", so without more information, such as the equivalence relation, "<math>\lesssim</math>" cannot be reconstructed from "<". Possible preorders include the following:
Mathematics is the language used for compact description of the order in nature, especially the laws of physics. This was noted and advocated by [[Pythagoras]],<ref name="dijksterhuis1986">{{harvnb|Dijksterhuis|1986}}</ref> [[Plato]],<ref name="mastin2010-plato">"Although usually remembered today as a philosopher, Plato was also one of ancient Greece's most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BC, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: 'Let no-one ignorant of geometry enter here.'" {{harv|Mastin|2010}}</ref> [[Galileo]],<ref name="toraldodifrancia1976p10-galileo">"Philosophy is written in that great book which ever lies before our eyes. I mean the universe, but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth." – [[Galileo]] (1623), ''[[The Assayer]]'', as quoted in {{harvnb|Toraldo Di Francia|1976|p=10}}</ref> and [[Isaac Newton|Newton]].
*Define ''a'' ≤ ''b'' as ''a'' < ''b'' or ''a'' = ''b'' (i.e., take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations <math>\lesssim</math> and ~.
*Define ''a'' <math>\lesssim</math> ''b'' as "not ''b'' < ''a''" (i.e., take the inverse complement of the relation), which corresponds to defining ''a'' ~ ''b'' as "neither ''a'' < ''b'' nor ''b'' < ''a''"; these relations <math>\lesssim</math> and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a [[strict weak order]]. The resulting preorder is [[total relation|total]], that is, a [[total preorder]].


Physics theories use mathematics<ref name="applicationsofmathematics">{{cite web |url=http://www.math.niu.edu/~rusin/known-math/index/tour_sci.html |title=Applications of Mathematics to the Sciences |publisher=Math.niu.edu |date=25 January 2000 |accessdate=30 January 2012}}</ref> to obtain order and provide precise formulas, [[analytic solution|precise]] or [[simulation#Computer simulation|estimated]] solutions, quantitative results and predictions. Experiment results in physics are numerical measurements. Technologies based on mathematics, like [[scientific computing|computation]] have made [[computational physics]] an active area of research.
==Number of preorders==
{{Number of relations}}


[[File:Mathematical Physics and other sciences.png|thumb|The distinction between mathematics and physics is clear-cut, but not always obvious, especially in mathematical physics.]]
As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
*for n=3:
**1 partition of 3, giving 1 preorder
**3 partitions of 2+1, giving 3 × 3 = 9 preorders
**1 partition of 1+1+1, giving 19 preorders
:i.e. together 29 preorders.
*for n=4:
**1 partition of 4, giving 1 preorder
**7 partitions with two classes (4  of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders
**6 partitions of 2+1+1, giving 6 × 19 = 114 preorders
**1 partition of 1+1+1+1, giving 219 preorders
:i.e. together 355 preorders.


[[Ontology]] is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while mathematical statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data.
==Interval==
For ''a'' <math>\lesssim</math> ''b'', the [[interval (mathematics)|interval]] [''a'',''b''] is the set of points ''x'' satisfying ''a'' <math>\lesssim</math> ''x'' and ''x'' <math>\lesssim</math> ''b'', also written ''a'' <math>\lesssim</math> ''x'' <math>\lesssim</math> ''b''. It contains at least the points ''a'' and ''b''. One may choose to extend the definition to all pairs (''a'',''b''). The extra intervals are all empty.


The distinction is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical.<ref name="jmp-def">{{cite web | url=http://www.researchgate.net/journal/0022-2488_Journal_of_Mathematical_Physics | title=Journal of Mathematical Physics | publisher=ResearchGate | accessdate=31 March 2014|quote=mathematical physics &mdash; that is, the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.}}</ref> The problems in this field start with a "[[Boundary condition|mathematical model of a physical situation]]" and a "mathematical description of a physical law". Every mathematical statement used for solution has a hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for.
Using the corresponding strict relation "<", one can also define the interval (''a'',''b'') as the set of points ''x'' satisfying ''a'' < ''x'' and ''x'' < ''b'', also written ''a'' < ''x'' < ''b''. An open interval may be empty even if ''a'' < ''b''.


Physics is a branch of [[fundamental science]], not [[practical science]].<ref name="aaas1917p645">American Association for the Advancement of Science, Science. 1917. Page 645</ref> Physics is also called "the fundamental science" because the subject of study of all branches of [[natural science]] like chemistry, astronomy, geology and biology are constrained by laws of physics,<ref name="feynmanleightonsands1963v1ch3">{{harvnb|Feynman|Leighton|Sands|1963|loc=Chapter 3: "The Relation of Physics to Other Sciences"}}; see also [[reductionism]] and [[special sciences]]</ref> similar to how chemistry is often called [[the central science]] because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and [[chemical reaction|reactions]] of matter (chemistry's focus on the atomic scale [[Difference between chemistry and physics|distinguishes it from physics]]). Structures are formed because particles exert electrical forces on each other, properties include physical characteristics of given substances, and reactions are bound by laws of physics, like conservation of energy, mass and charge.
Also [''a'',''b'') and (''a'',''b''] can be defined similarly.
 
Physics is applied in industries like engineering and medicine. <!--Please add a phrase about how physics first caused industrial revolution, with reliable source-->
 
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===Application and influence===
{{Main|Applied physics}}
[[File:Archimedes-screw one-screw-threads with-ball 3D-view animated small.gif|thumb|150px|[[Archimedes' screw]], a [[simple machine]] for lifting]]
[[File:IMG 1729 Gemaal met schroef van Archimedes bij Kinderdijk.JPG|thumb|150px|The application of physical laws in lifting liquids]]
 
[[Applied physics]] is a general term for physics research which is intended for a particular [[Utility|use]]. An applied physics [[curriculum]] usually contains a few classes in an applied discipline, like geology or electrical engineering. It usually differs from [[engineering]] in that an applied physicist may not be designing something in particular, but rather is using physics or conducting physics research with the aim of developing new technologies or solving a problem.
 
The approach is similar to that of [[applied mathematics]]. Applied physicists can also be interested in the use of physics for scientific research. For instance, people working on [[accelerator physics]] might seek to build better [[particle detector]]s for research in theoretical physics.
 
Physics is used heavily in [[engineering]]. For example, [[statics]], a subfield of [[mechanics]], is used in the building of [[bridge]]s and other static structures. The understanding and use of [[acoustics]] results in sound control and better concert halls; similarly, the use of [[optics]] creates better optical devices. An understanding of physics makes for more realistic [[flight simulator]]s, [[video game]]s, and [[Film|movies]], and is often critical in [[forensic]] investigations.
 
With the [[Uniformitarianism (science)|standard consensus]] that the [[Scientific law|laws]] of physics are universal and do not change with time, physics can be used to study things that would ordinarily be mired in [[uncertainty]]. For example, in the [[History of Earth#Origin of the Earth's core and first atmosphere|study of the origin of the earth]], one can reasonably model earth's [[mass]], [[temperature]], and rate of [[rotation]], as a function of time allowing one to extrapolate forward and backward in time and so predict prior and future conditions. It also allows for simulations in engineering which drastically speed up the development of a new technology.
 
But there is also considerable [[interdisciplinarity]] in the physicist's methods, so many other important fields are influenced by physics (e.g., the fields of [[econophysics]] and sociophysics).
 
==Research==
 
===Scientific method===
Physicists use [[scientific method|the scientific method]] to test the validity of a [[physical theory]], using a methodical approach to compare the implications of the theory in question with the associated conclusions drawn from [[experiment]]s and observations conducted to test it. Experiments and observations are collected and compared with the predictions and hypotheses made by a theory, thus aiding in the determination or the validity/invalidity of the theory.
 
A [[scientific law]] is a concise verbal or mathematical statement of a relation which expresses a fundamental principle of some theory, such as Newton's law of universal gravitation.<ref name="honderich1995pp474-476">{{harvnb|Honderich|1995|pp=474–476}}</ref>
 
===Theory and experiment===
{{Main|Theoretical physics|Experimental physics}}
[[File:Astronaut-EVA.jpg|thumb|right|The [[astronaut]] and [[Earth]] are both in [[free-fall]]]]
[[File:Lightning in Arlington.jpg|thumb|left|[[Lightning]] is an [[electric current]]]]
 
Theorists seek to develop [[mathematical model]]s that both agree with existing experiments and successfully predict future experimental results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although [[theory]] and [[experiment]] are developed separately, they are strongly dependent upon each other. Progress in physics frequently comes about when [[experimentalist]]s make a discovery that existing theories cannot explain, or when new theories generate experimentally testable [[prediction]]s, which inspire new experiments.
 
[[Physicist]]s who work at the interplay of [[theory]] and [[experiment]] are called [[Phenomenology (science)|phenomenologists]]. Phenomenologists look at the complex phenomena observed in experiment and work to relate them to fundamental theory.
 
Theoretical physics has historically taken inspiration from philosophy; [[electromagnetism]] was unified this way.{{efn|See, for example, the influence of [[Immanuel Kant|Kant]] and [[Johann Wilhelm Ritter|Ritter]] on [[Hans Christian Ørsted|Oersted]].}} Beyond the known universe, the field of theoretical physics also deals with hypothetical issues,{{efn|Concepts which are denoted ''hypothetical'' can change with time. For example, the [[atom]] of nineteenth century physics was denigrated by some, including [[Ernst Mach]]'s critique of [[Ludwig Boltzmann]]'s formulation of [[statistical mechanics]]. By the end of [[World War II]], the atom was no longer deemed hypothetical.}} such as [[Many-worlds interpretation|parallel universes]], a [[multiverse]], and [[higher dimension]]s. Theorists invoke these ideas in hopes of solving particular problems with existing theories. They then explore the consequences of these ideas and work toward making testable predictions.
 
[[Experiment]]al physics expands, and is expanded by, [[engineering]] and [[technology]]. Experimental physicists involved in [[basic research]] design and perform experiments with equipment such as [[particle accelerator]]s and [[laser]]s, whereas those involved in [[applied research]] often work in industry developing technologies such as [[MRI|magnetic resonance imaging (MRI)]] and [[transistor]]s. [[Richard Feynman|Feynman]] has noted that experimentalists may seek areas which are not well-explored by theorists.<ref name="feynman1965p157-experiment">"In fact experimenters have a certain individual character. They ... very often do their experiments in a region in which people know the theorist has not made any guesses." {{harv|Feynman|1965|p=157}}</ref>
 
{{clear}}
 
===Scope and aims===
[[File:Acceleration components.JPG|thumb|left|Physics involves modeling the natural world with theory, usually quantitative. Here, the path of a particle is modeled with the mathematics of [[calculus]] to explain its behavior: the purview of the branch of physics known as [[mechanics]].]]
 
Physics covers a wide range of [[phenomenon|phenomena]], from [[elementary particle]]s (such as quarks, neutrinos, and electrons) to the largest [[superclusters]] of galaxies. Included in these phenomena are the most basic objects composing all other things. Therefore physics is sometimes called the "[[fundamental science]]".<ref name="feynmanleightonsands1963v1ch3" /> Physics aims to describe the various phenomena that occur in nature in terms of simpler phenomena. Thus, physics aims to both connect the things observable to humans to [[root cause]]s, and then connect these causes together.
 
For example, the [[History of China|ancient Chinese]] observed that certain rocks ([[lodestone]]) were attracted to one another by some invisible force. This effect was later called [[magnetism]], and was first rigorously studied in the 17th century. A little earlier than the Chinese, the [[Ancient Greece|ancient Greeks]] knew of other objects such as [[amber]], that when rubbed with fur would cause a similar invisible attraction between the two. This was also first studied rigorously in the 17th century, and came to be called [[electricity]]. Thus, physics had come to understand two observations of nature in terms of some root cause (electricity and magnetism). However, further work in the 19th century revealed that these two forces were just two different aspects of one force—[[electromagnetism]]. This process of "unifying" forces continues today, and electromagnetism and the [[weak nuclear force]] are now considered to be two aspects of the [[electroweak interaction]]. Physics hopes to find an ultimate reason ([[Theory of Everything]]) for why nature is as it is (see section ''[[#Current research|Current research]]'' below for more information).
 
===Research fields===
Contemporary research in physics can be broadly divided into [[condensed matter physics]]; [[atomic, molecular, and optical physics]]; [[particle physics]]; [[astrophysics]]; [[geophysics]] and [[biophysics]]. Some physics departments also support [[physics education research]] and [[physics outreach]].
 
Since the 20th century, the individual fields of physics have become increasingly [[specialization of knowledge|specialized]], and today most physicists work in a single field for their entire careers. "Universalists" such as [[Albert Einstein]] (1879–1955) and [[Lev Landau]] (1908–1968), who worked in multiple fields of physics, are now very rare.{{efn|Yet, universalism is encouraged in the culture of physics. For example, the [[World Wide Web]], which was innovated at [[CERN]] by [[Tim Berners-Lee]], was created in service to the computer infrastructure of CERN, and was/is intended for use by physicists worldwide. The same might be said for [[arXiv.org]]}}
 
The major fields of physics, along with their subfields and the theories they employ, are shown in the following table.
{{Subfields of physics}}
 
====Condensed matter====
{{Main|Condensed matter physics}}
[[File:Bose Einstein condensate.png|right|thumb|350px|Velocity-distribution data of a gas of [[rubidium]] atoms, confirming the discovery of a new phase of matter, the [[Bose–Einstein condensate]]]]
 
[[Condensed matter physics]] is the field of physics that deals with the macroscopic physical properties of matter.<ref name="taylorheinonen2002">{{harvnb|Taylor|Heinonen|2002}}</ref> In particular, it is concerned with the "condensed" [[phase (matter)|phases]] that appear whenever the number of particles in a system is extremely large and the interactions between them are strong.<ref name=cohen2008>{{harvnb|Cohen|2008}}</ref>
 
The most familiar examples of condensed phases are [[Solid-state physics|solids]] and [[liquid]]s, which arise from the bonding by way of the [[electromagnetic force]] between [[atom]]s.<ref name="moore2011">{{harvnb|Moore|2011|pp=255–258}}</ref> More exotic condensed phases include the [[superfluid]]<ref name="leggett1999">{{harvnb|Leggett|1999}}</ref> and the [[Bose–Einstein condensate]]<ref name="levy2001">{{harvnb|Levy|2001}}</ref> found in certain atomic systems at very low [[temperature]], the [[superconductivity|superconducting]] phase exhibited by [[conduction electron]]s in certain materials,<ref name=stajiccoontzosborne2011>{{harvnb|Stajic|Coontz|Osborne|2011}}</ref> and the [[ferromagnet]]ic and [[antiferromagnet]]ic phases of [[spin (physics)|spins]] on [[crystal lattice|atomic lattices]].<ref name="mattis2006">{{harvnb|Mattis|2006}}</ref>
 
Condensed matter physics is the largest field of contemporary physics. Historically, condensed matter physics grew out of [[solid-state physics]], which is now considered one of its main subfields.<ref name="aps-dcmp">{{cite web | url=http://www.aps.org/units/dcmp/history.cfm | title=History of Condensed Matter Physics | publisher=[[American Physical Society]] | accessdate=31 March 2014}}</ref> The term ''condensed matter physics'' was apparently coined by [[Philip Warren Anderson|Philip Anderson]] when he renamed his research group—previously ''solid-state theory''—in 1967.<ref name="princeton-anderson">{{cite web|title=Philip Anderson|url=http://www.princeton.edu/physics/people/display_person.xml?netid=pwa&display=faculty|work=Physics Faculty|publisher=Princeton University|accessdate=15 October 2012}}</ref> In 1978, the Division of Solid State Physics of the [[American Physical Society]] was renamed as the Division of Condensed Matter Physics.<ref name="aps-dcmp" /> Condensed matter physics has a large overlap with [[chemistry]], [[materials science]], [[nanotechnology]] and [[engineering]].<ref name="cohen2008" />
 
====Atomic, molecular, and optical physics====
{{Main|Atomic, molecular, and optical physics}}
 
[[Atom]]ic, [[Molecule|molecular]], and [[Optics|optical]] physics (AMO) is the study of [[matter]]–matter and [[light]]–matter interactions on the scale of single [[atom]]s and molecules. The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of their relevant [[energy]] scales. All three areas include both [[classical physics|classical]], semi-classical and [[quantum physics|quantum]] treatments; they can treat their subject from a microscopic view (in contrast to a macroscopic view).
 
[[Atomic physics]] studies the [[electron]] shells of [[atom]]s. Current research focuses on activities in quantum control, cooling and trapping of atoms and ions,<ref>For example, AMO research groups at {{cite web |url=http://web.mit.edu/physics/research/abcp/areas.html#amo |title=MIT AMO Group |accessdate=21 February 2014}}</ref><ref>{{cite web |url=http://physics.korea.ac.kr/research/research_amo.php |title=Korea University, Physics AMO Group |accessdate=21 February 2014}}</ref><ref>{{cite web |url=http://phys.au.dk/forskning/forskningsomraader/amo/ |title=Aarhus Universitet, AMO Group |accessdate=21 February 2014}}</ref> low-temperature collision dynamics and the effects of electron correlation on structure and dynamics. Atomic physics is influenced by the [[Atomic nucleus|nucleus]] (see, e.g., [[hyperfine splitting]]), but intra-nuclear phenomena such as [[nuclear fission|fission]] and [[nuclear fusion|fusion]] are considered part of [[high-energy physics]].
 
[[Molecular physics]] focuses on multi-atomic structures and their internal and external interactions with matter and light. [[Optical physics]] is distinct from [[optics]] in that it tends to focus not on the control of classical light fields by macroscopic objects but on the fundamental properties of [[optical field]]s and their interactions with matter in the microscopic realm.
 
====High-energy physics (particle physics) and nuclear physics====
{{anchor|High energy physics (particle physics) and nuclear physics}}
{{Main|Particle physics|Nuclear physics}}
[[File:CMS Higgs-event.jpg|thumb|A simulated event in the CMS detector of the [[Large Hadron Collider]], featuring a possible appearance of the [[Higgs boson]].]]
 
[[Particle physics]] is the study of the [[elementary particle|elementary]] constituents of [[matter]] and [[energy]] and the [[Fundamental interaction|interactions]] between them.<ref name="aps-dpf">{{cite web|title=Division of Particles & Fields|url=http://www.aps.org/units/dpf/index.cfm|publisher=American Physical Society|accessdate=18 October 2012}}</ref> In addition, particle physicists design and develop the high energy [[particle accelerator|accelerators]],<ref name="halpern2010">{{harvnb|Halpern|2010}}</ref> [[Particle detector|detectors]],<ref name="grupen1999">{{harvnb|Grupen|1999}}</ref> and [[Computational particle physics|computer programs]]<ref name="walsh2012">{{harvnb|Walsh|2012}}</ref> necessary for this research. The field is also called "high-energy physics" because many elementary particles do not occur naturally but are created only during high-energy [[collision]]s of other particles.<ref name="iop-hepp">{{cite web|title=High Energy Particle Physics Group|url=http://www.iop.org/activity/groups/subject/hepp/index.html|publisher=Institute of Physics|accessdate=18 October 2012}}</ref>
 
Currently, the interactions of elementary particles and [[Field (physics)|fields]] are described by the [[Standard Model]].<ref name="oerter2006">{{harvnb|Oerter|2006}}</ref> The model accounts for the 12 known particles of matter ([[quark]]s and [[lepton]]s) that interact via the [[strong nuclear force|strong]], [[weak nuclear force|weak]], and [[electromagnetism|electromagnetic]] [[fundamental force]]s.<ref name="oerter2006" /> Dynamics are described in terms of matter particles exchanging [[gauge boson]]s ([[gluon]]s, [[W and Z bosons]], and [[photon]]s, respectively).<ref name="gribbin1998">{{harvnb|Gribbin|Gribbin|Gribbin|1998}}</ref> The Standard Model also predicts a particle known as the [[Higgs boson]].<ref name="oerter2006" /> In July 2012 [[CERN]], the European laboratory for particle physics, announced the detection of a particle consistent with the Higgs boson,<ref name="eonr-higgs">{{cite web|title=CERN experiments observe particle consistent with long-sought Higgs boson|url=http://press-archived.web.cern.ch/press-archived/PressReleases/Releases2012/PR17.12E.html|publisher=European Organization for Nuclear Research|accessdate=18 October 2012|date=4 July 2012}}</ref> an integral part of a [[Higgs mechanism]].
 
[[Nuclear physics]] is the field of physics that studies the constituents and interactions of [[atomic nuclei]].  The most commonly known applications of nuclear physics are [[nuclear power]] generation and [[nuclear weapons]] technology, but the research has provided application in many fields, including those in [[nuclear medicine]] and [[magnetic resonance imaging]], [[ion implantation]] in [[materials engineering]], and [[radiocarbon dating]] in [[geology]] and [[archaeology]].
 
====Astrophysics====
{{Main|Astrophysics|Physical cosmology}}
[[File:Hubble ultra deep field high rez edit1.jpg|thumb|left|The deepest visible-light image of the [[universe]], the [[Hubble Ultra Deep Field]]]]
 
[[Astrophysics]] and [[astronomy]] are the application of the theories and methods of physics to the study of [[stellar structure]], [[stellar evolution]], the origin of the [[solar system]], and related problems of [[Physical cosmology|cosmology]]. Because astrophysics is a broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.
 
The discovery by [[Karl Jansky]] in 1931 that radio signals were emitted by celestial bodies initiated the science of [[radio astronomy]]. Most recently, the frontiers of astronomy have been expanded by space exploration. Perturbations and interference from the earth's atmosphere make space-based observations necessary for [[infrared astronomy|infrared]], [[ultraviolet astronomy|ultraviolet]], [[gamma-ray astronomy|gamma-ray]], and [[X-ray astronomy]].
 
[[Physical cosmology]] is the study of the formation and evolution of the universe on its largest scales. Albert Einstein's theory of relativity plays a central role in all modern cosmological theories. In the early 20th century, [[Edwin Hubble|Hubble]]'s discovery that the universe is expanding, as shown by the [[Hubble diagram]], prompted rival explanations known as the [[steady state theory|steady state]] universe and the [[Big Bang]].
 
The Big Bang was confirmed by the success of [[Big Bang nucleosynthesis]] and the discovery of the [[cosmic microwave background]] in 1964. The Big Bang model rests on two theoretical pillars: Albert Einstein's general relativity and the [[cosmological principle]]. Cosmologists have recently established the [[Lambda-CDM model|ΛCDM model]] of the evolution of the universe, which includes [[cosmic inflation]], [[dark energy]], and [[dark matter]].
 
Numerous possibilities and discoveries are anticipated to emerge from new data from the [[Fermi Gamma-ray Space Telescope]] over the upcoming decade and vastly revise or clarify existing models of the [[universe]].<ref name="nasa-glast">{{cite web |url=http://www.nasa.gov/mission_pages/GLAST/main/questions_answers.html |title=NASA – Q&A on the GLAST Mission |accessdate=29 April 2009 |work=Nasa: Fermi Gamma-ray Space Telescope |publisher=[[NASA]] |date=28 August 2008}}</ref><ref>See also [http://www.nasa.gov/mission_pages/GLAST/science/index.html Nasa – Fermi Science] and [http://www.nasa.gov/mission_pages/GLAST/science/unidentified_sources.html NASA – Scientists Predict Major Discoveries for GLAST].</ref> In particular, the potential for a tremendous discovery surrounding dark matter is possible over the next several years.<ref name="nasa-glast-darkmatter">{{cite web |url=http://www.nasa.gov/mission_pages/GLAST/science/dark_matter.html |title=Dark Matter |publisher=Nasa.gov |date=28 August 2008 |accessdate=30 January 2012}}</ref> Fermi will search for evidence that dark matter is composed of [[weakly interacting massive particle]]s, complementing similar experiments with the [[Large Hadron Collider]] and other underground detectors.
 
[[IBEX]] is already yielding new [[astrophysical]] discoveries: "No one knows what is creating the [[energetic neutral atom|ENA (energetic neutral atoms)]] ribbon" along the [[termination shock]] of the [[solar wind]], "but everyone agrees that it means the textbook picture of the [[heliosphere]] — in which the solar system's enveloping pocket filled with the solar wind's charged particles is plowing through the onrushing 'galactic wind' of the interstellar medium in the shape of a comet — is wrong."<ref name="kerr2009">{{harvnb|Kerr|2009}}</ref>
 
==Current research==
{{Further2|[[List of unsolved problems in physics]]}}
[[File:Feynman'sDiagram.JPG|thumb|right|[[Feynman diagram]] signed by [[R.P. Feynman]]]]
[[File:Meissner effect p1390048.jpg|thumb|right|A typical event described by physics: a [[magnet]] levitating above a [[superconductor]] demonstrates the [[Meissner effect]].]]
 
Research in physics is continually progressing on a large number of fronts.
 
In condensed matter physics, an important unsolved theoretical problem is that of [[high-temperature superconductivity]]. Many condensed matter experiments are aiming to fabricate workable [[spintronics]] and [[quantum computer]]s.
 
In particle physics, the first pieces of experimental evidence for physics beyond the [[Standard Model]] have begun to appear. Foremost among these are indications that [[neutrino]]s have non-zero [[mass]]. These experimental results appear to have solved the long-standing [[solar neutrino problem]], and the physics of massive neutrinos remains an area of active theoretical and experimental research. [[Particle accelerator]]s have begun probing energy scales in the [[TeV]] range, in which experimentalists are hoping to find evidence for the [[Higgs boson]] and [[supersymmetry|supersymmetric particles]].<ref>{{harvnb|DØ  Collaboration|2007}} finds a mass of 5.774 GeV for the <math>\Xi_{b}^{-}</math></ref>
 
Theoretical attempts to unify [[quantum mechanics]] and [[general relativity]] into a single theory of [[quantum gravity]], a program ongoing for over half a century, have not yet been decisively resolved. The current leading candidates are [[M-theory]], [[superstring theory]] and [[loop quantum gravity]].
 
Many [[astronomical]] and [[physical cosmology|cosmological]] phenomena have yet to be satisfactorily explained, including the existence of [[GZK paradox|ultra-high energy cosmic rays]], the [[baryon asymmetry]], the [[accelerating universe|acceleration of the universe]] and the [[galaxy rotation problem|anomalous rotation rates of galaxies]].
 
Although much progress has been made in high-energy, [[quantum]], and astronomical physics, many everyday phenomena involving [[complex systems|complexity]],<ref name="nrc1997v9p161">{{harvnb|National Research Council|Committee on Technology for Future Naval Forces|1997|p=161}}</ref> [[Chaos theory|chaos]],<ref name="kellert1993p32">{{harvnb|Kellert|1993|p=32}}</ref> or [[turbulence]]<ref name="burchard2002p2">{{harvnb|Burchard|2002|p=2}}</ref> are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics remain unsolved; examples include the formation of sandpiles, nodes in trickling [[water]], the shape of water [[droplet]]s, mechanisms of [[surface tension]] [[catastrophe theory|catastrophes]], and self-sorting in shaken heterogeneous collections.{{citation needed|time=2010-11-03|date=November 2010}}
 
These complex phenomena have received growing attention since the 1970s for several reasons, including the availability of modern [[mathematical]] methods and [[computers]], which enabled [[complex systems]] to be modeled in new ways. Complex physics has become part of increasingly [[interdisciplinary]] research, as exemplified by the study of [[turbulence]] in [[aerodynamics]] and the observation of [[pattern formation]] in [[biological]] systems. In 1932, [[Horace Lamb]] said:<ref name="goldstein1969">{{harvnb|Goldstein|1969}}</ref>
{{Quote|text=I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.|sign=[[Horace Lamb]]|source=Annual Reviews in Fluid Mechanics}}


==See also==
==See also==
{{Portal|Physics}}
*[[partially ordered set|partial order]] - preorder that is [[antisymmetric relation|antisymmetric]]
{{Wikipedia books}}
*[[equivalence relation]] - preorder that is [[Symmetric relation|symmetric]]
 
*[[Strict weak ordering#Total preorders|total preorder]] - preorder that is [[Total relation|total]]
{{Col-begin}}{{Col-break}}
*[[total order]] - preorder that is antisymmetric and total
;General
*[[directed set]]
* [[Glossary of classical physics]]
*[[category of preordered sets]]
* [[Glossary of physics]]
*[[prewellordering]]
* [[Index of physics articles]]
*[[preordered class]]
* [[List of elementary physics formulae]], [[Elementary physics formulae]]
*[[Well-quasi-ordering]]
* [[List of important publications in physics]]
*[[Newman's lemma]]
* [[List of physicists]]
* [[List of physics concepts in primary and secondary education curricula]]
* [[Physics Outreach]]
* [[Perfection#Physics and chemistry|Perfection in physics and chemistry]]
* [[Relationship between mathematics and physics]]
* [[Timeline of developments in theoretical physics]]
* [[Timeline of fundamental physics discoveries]]
 
;Main branches
* [[Classical mechanics|Classical Mechanics]]
* [[Electromagnetism|Electricity and Magnetism]]
* [[Modern physics|Modern Physics]]
* [[Optics]]
* [[Thermodynamics]]
 
{{Col-break}}
 
;Related fields
* [[Astronomy]]
* [[Chemistry]]
* [[Engineering]]
* [[Mathematics]]
* [[Quantum Mechanics]]
* [[Science]]
* [[Cosmology]]
 
;Interdisciplinary fields incorporating physics
* [[Acoustics]]
* [[Biophysics]]
* [[Econophysics]]
* [[Geophysics]]
* [[Nanotechnology]]
* [[Neurophysics]]
* [[Psychophysics]]
{{Col-end}}
 
{{Portal bar|Physics|Cosmology}}
 
==Notes==
{{notelist}}


==References==
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</div>
 
==External links==
{{Wiktionary|physics}}
{{Wikibooks|Physics}}
{{Wikibooks|Physics Study Guide}}
{{Wikibooks|FHSST Physics}}
{{Wikisource portal|Physics}}
{{Wikiversity|Category:Physics|Physics}}
 
'''General'''
<!--Please do not post more links here, they will be taken down as link spam!!-->
* [http://www.scholarpedia.org/article/Encyclopedia_of_physics Encyclopedia of Physics] at [[Scholarpedia]]
* de Haas, Paul, {{Wayback|url=http://home.tiscali.nl/physis/HistoricPaper/|title=Historic Papers in Physics (20th Century)|date=20090826083339}}
* [http://www.physicscentral.com/ PhysicsCentral] – Web portal run by the [http://www.aps.org/ American Physical Society]
* [http://www.physics.org/ Physics.org] – Web portal run by the [http://www.iop.org/ Institute of Physics]
* [http://musr.physics.ubc.ca/~jess/hr/skept/ ''The Skeptic's Guide to Physics'']
* [http://math.ucr.edu/home/baez/physics/ Usenet Physics FAQ] – A FAQ compiled by sci.physics and other physics newsgroups
* [http://nobelprize.org/nobel_prizes/physics/ Website of the Nobel Prize in physics]
* [http://scienceworld.wolfram.com/physics/ World of Physics]  An online encyclopedic dictionary of physics
* [http://www.nature.com/naturephysics ''Nature'': Physics]
* [http://physics.aps.org/ Physics] announced 17 July 2008 by the [[American Physical Society]]
* {{dmoz|/Science/Physics/Publications/|Physics/Publications}}
* [http://physicsworld.com/ Physicsworld.com] – News website from [http://publishing.iop.org/ Institute of Physics Publishing]
* [http://physlib.com/ Physics Central] – includes articles on astronomy, particle physics, and mathematics.
* [http://www.vega.org.uk/ The Vega Science Trust] – science videos, including physics
* [https://archive.org/details/JustinMorganPhysicsLightningTour/ Video: Physics "Lightning" Tour with Justin Morgan]
* [http://www.learner.org/resources/series42.html 52-part video course: The Mechanical Universe...and Beyond] Note: also available at [https://video.google.com/videoplay?docid=-6774539130229106025 01 – Introduction]{{dead link|date=September 2014}} at [[Google Videos]]
* [http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html HyperPhysics website] – [[HyperPhysics]], a physics and astronomy mind-map from [[Georgia State University]]
 
'''Organizations'''
* [http://www.aip.org/index.html AIP.org] – Website of the [[American Institute of Physics]]
* [http://www.aps.org/ APS.org] – Website of the [[American Physical Society]]
* [http://www.iop.org/ IOP.org] – Website of the [[Institute of Physics]]
* [http://planetphysics.org/ PlanetPhysics.org]
* [http://www.royalsoc.ac.uk/ Royal Society] – Although not exclusively a physics institution, it has a strong history of physics
* [http://www.spsnational.org/ SPS National] – Website of the [[Society of Physics Students]]
 
{{Fundamental interactions}}
{{Branches of physics}}
{{Natural science}}


<!--The parental lineage categories are incomplete without physics listed in them. Physics was a blatant commission, and makes those lists look rather unprofessional, considering the other fields are listed.-->
[[Category:Order theory]]
[[Category:Mathematical relations]]


[[Category:Physics| ]]
[[cs:Kvaziuspořádání]]
[[da:Præordning]]
[[de:Quasiordnung]]
[[es:Conjunto preordenado]]
[[fr:Pré-ordre]]
[[it:Preordine]]
[[he:קדם סדר]]
[[pl:Praporządek]]
[[ru:Предпорядок]]
[[sk:Kváziusporiadanie]]
[[uk:Передпорядок]]
[[zh:预序关系]]

Revision as of 00:13, 9 August 2014

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In mathematics, especially in order theory, a preorder or quasi-order is a binary relation that is reflexive and transitive. All partial orders and equivalence relations are preorders, but preorders are more general.

The name 'preorder' comes from the idea that preorders are 'almost' (partial) orders, but not quite; they're neither anti-symmetric nor symmetric. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not anti-symmetric, some of the ordinary intuition that a student may have with regards to the symbol ≤ may not apply. On the other hand, a pre-order can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worth-while, depending on the problem domain being studied.

In words, when a ≤ b, one may say that b covers a or that b precedes a, or that b reduces to a. Occasionally, the notation ← or ≲ is used instead of ≤.

To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs may be cyclic graphs: preorders may have cycles in them. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder may have many disconnected components. The diamond lemma is an important result for certain kinds of preorders.

Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)
if a ≤ b and b ≤ c then a ≤ c (transitivity)

Note that an alternate definition of preorder requires the relation to be irreflexive. However, as this article is examining preorders as a logical extension of non-strict partial orders, the current definition is more intuitive.

A set that is equipped with a preorder is called a preordered set (or proset).

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.

On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.

A preorder which is preserved in all contexts (i.e. respected by all functions on P) is called a precongruence. A precongruence which is also symmetric (i.e. is an equivalence relation) is a congruence relation.

Equivalently, a preordered set P can be defined as a category with objects the elements of P, and each hom-set having at most one element (one for objects which are related, zero otherwise).

Alternately, a preordered set can be understood as an enriched category, enriched over the category 2 = (0→1).

Examples

  • The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, where x ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for every pair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partially ordered sets (preorders satisfying an additional anti-symmetry property).
  • Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not 1-to-1.
  • A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without losing important features.
  • The relation defined by x≤y iff f(x)≤f(y), where f is a function into some preorder.
  • The relation defined by x≤y iff there exists some injection from x to y. Injection may be replaced by surjection, or any type of structure-preserving function, such as ring homomorphism, or permutation.
  • The embedding relation for countable total orderings.
  • The graph-minor relation in graph theory.
  • A category with at most one morphism between any pair of objects is a preorder. Such categories are called thin. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

Example of a total preorder:

Uses

Preorders play a pivotal role in several situations:

Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x to y.

Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲ a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.)

Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y] if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y. By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.

Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).

For a preorder "≲", a relation "<" can be defined as a < b if and only if (a ≲ b and not b ≲ a), or equivalently, using the equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial order can be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).

(Alternatively, for a preorder "≲", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before.)

Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation "≲"; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.

Note that with this construction multiple preorders "≲" can give the same relation "<", so without more information, such as the equivalence relation, "≲" cannot be reconstructed from "<". Possible preorders include the following:

  • Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations ≲ and ~.
  • Define a ≲ b as "not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a ~ b as "neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

Number of preorders

Template:Number of relations

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:

  • for n=3:
    • 1 partition of 3, giving 1 preorder
    • 3 partitions of 2+1, giving 3 × 3 = 9 preorders
    • 1 partition of 1+1+1, giving 19 preorders
i.e. together 29 preorders.
  • for n=4:
    • 1 partition of 4, giving 1 preorder
    • 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders
    • 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders
    • 1 partition of 1+1+1+1, giving 219 preorders
i.e. together 355 preorders.

Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains at least the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.

Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

Also [a,b) and (a,b] can be defined similarly.

See also

References

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