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| In the [[mathematics|mathematical]] areas of [[order theory|order]] and [[lattice theory]], the '''Knaster–Tarski theorem''', named after [[Bronisław Knaster]] and [[Alfred Tarski]], states the following:
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| :''Let L be a [[complete lattice]] and let f : L → L be an [[monotonic function#Monotonicity_in_order_theory|order-preserving]] [[function (mathematics)|function]]. Then the [[Set (mathematics)|set]] of [[fixed point (mathematics)|fixed point]]s of f in L is also a complete lattice.''
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| It was Tarski who stated the result in its most general form,<ref>{{cite journal | author=Alfred Tarski | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044538 | title=A lattice-theoretical fixpoint theorem and its applications | journal = Pacific Journal of Mathematics | volume=5:2 | year=1955 | pages=285–309}}</ref> and so the theorem is often known as '''Tarski's fixed point theorem'''. Some time earlier, Knaster and Tarski established the result for the special case where ''L'' is the lattice of subsets of a set, the [[power set]] lattice.<ref>{{cite journal | author=B. Knaster | title=Un théorème sur les fonctions d'ensembles | journal=Ann. Soc. Polon. Math. | year=1928 | volume=6 | pages=133–134}} With A. Tarski.</ref>
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| The theorem has important applications in [[formal semantics of programming languages]] and [[abstract interpretation]].
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| A kind of converse of this theorem was proved by [[Anne C. Davis]]: If every order preserving function ''f : L → L'' on a [[lattice (order)|lattice]] ''L'' has a fixed point, then ''L'' is a complete lattice.<ref>{{cite journal | author=Anne C. Davis | url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044539 | title=A characterization of complete lattices | journal=Pacific J. Math. | year=1955 | volume=5 | pages=311–319}}</ref>
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| ==Consequences: least and greatest fixed points==
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| Since complete lattices cannot be [[empty set|empty]], the theorem in particular guarantees the existence of at least one fixed point of ''f'', and even the existence of a ''least'' (or ''greatest'') fixed point. In many practical cases, this is the most important implication of the theorem.
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| The [[least fixpoint]] of ''f'' is the least element ''x'' such that ''f''(''x'') = ''x'', or, equivalently, such that ''f''(''x'') ≤ ''x''; the [[duality (order theory)|dual]] holds for the [[greatest fixpoint]], the greatest element ''x'' such that ''f''(''x'') = ''x''.
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| If ''f''(lim ''x''<sub>''n''</sub>)=lim ''f''(''x''<sub>''n''</sub>) for all ascending sequences ''x''<sub>''n''</sub>, then the least fixpoint of ''f'' is lim ''f''<sup>''n''</sup>(0) where 0 is the least element of L, thus giving a more "constructive" version of the theorem. (See: [[Kleene fixed-point theorem]].) More generally, if ''f'' is monotonic, then the least fixpoint of ''f'' is the stationary limit of ''f''<sup>α</sup>(0), taking α over the [[ordinal number|ordinals]], where ''f''<sup>α</sup> is defined by [[transfinite induction]]: ''f''<sup>α+1</sup> = ''f'' ( ''f''<sup>α</sup>) and ''f''<sup>γ</sup> for a limit ordinal γ is the [[least upper bound]] of the ''f''<sup>β</sup> for all β ordinals less than γ. The dual theorem holds for the greatest fixpoint.
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| For example, in theoretical [[computer science]], [[least fixed point]]s of [[monotone function]]s are used to define [[program semantics]]. Often a more specialized version of the theorem is used, where ''L'' is assumed to be the lattice of all [[subset]]s of a certain set ordered by subset inclusion. This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function ''f''. [[Abstract interpretation]] makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoints. | |
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| Knaster–Tarski theorem can be used for a simple proof of [[Cantor–Bernstein–Schroeder theorem]].<ref>Example 3 in R. Uhl, "[http://mathworld.wolfram.com/TarskisFixedPointTheorem.html Tarski's Fixed Point Theorem]", from ''MathWorld''--a Wolfram Web Resource, created by Eric W. Weisstein.</ref>
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| ==Weaker versions of the theorem==
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| Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example:
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| :''Let L be a [[partially ordered set]] with the smallest element (bottom) and let f : L → L be an [[monotonic function|order-preserving]] [[function (mathematics)|function]]. Further, suppose there exists u in L such that f(u) ≤ u and that any chain in the subset {x in L : x ≤ f(x), x ≤ u} has supremum. Then f admits the least [[fixed point (mathematics)|fixed point]].''
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| This can be applied to obtain various theorems on [[invariant set]]s, e.g. the Ok's theorem:
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| :''For the monotone map F : P(X) → P(X) on the [[powerset|family]] of (closed) nonempty subsets of X the following are equivalent: (o) F admits A in P(X) s.t. <math>A \subseteq F(A)</math>, (i) F admits invariant set A in P(X) i.e. <math>A = F(A)</math>, (ii) F admits maximal invariant set A, (ii) F admits the greatest invariant set A.''
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| In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous (multivalued) [[iterated function system]]s. For weakly contractive iterated function systems [[Kantorovitch fixpoint theorem]] suffices.
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| Other applications of fixed point principles for ordered sets come from the theory of differential, integral and operator equations.
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| ==Proof==
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| Let's restate the theorem.
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| For a complete lattice <math>\langle L, \le \rangle</math> and a monotone function <math>f\colon L \rightarrow L</math> on ''L'', the set of all fixpoints of ''f'' is also a complete lattice <math>\langle P, \le \rangle</math>, with:
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| * <math>\bigvee P = \bigvee \{ x \in L \mid x \le f(x) \}</math> as the greatest fixpoint of ''f''
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| * <math>\bigwedge P = \bigwedge \{ x \in L \mid x \ge f(x) \}</math> as the least fixpoint of ''f''.
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| ''Proof.'' We begin by showing that ''P'' has least and greatest element. Let ''D'' = { ''x'' | ''x'' ≤ ''f(x)'' } and ''x'' ∈ ''D'' (we know that at least ''0<sub>L</sub>'' belongs to ''D''). Then because ''f'' is monotone we have ''f(x)'' ≤ ''f(f(x))'', that is ''f(x)'' ∈ ''D''.
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| Now let <math>u = \bigvee D</math>. Then ''x'' ≤ ''u'' and ''f(x)'' ≤ ''f(u)'', so ''x'' ≤ ''f(x)'' ≤ ''f(u)''. Therefore ''f(u)'' is an upper bound of ''D'', but ''u'' is the least upper bound, so ''u'' ≤ ''f(u)'', i.e. ''u'' ∈ ''D''. Then ''f(u)'' ∈ ''D'' and ''f(u)'' ≤ ''u'' from which follows ''f(u)'' = ''u''. Because every fixpoint is in ''D'' we have that ''u'' is the greatest fixpoint of ''f''.
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| The function ''f'' is monotone on the dual (complete) lattice <math>\langle L^{op}, \ge \rangle</math>. As we have just proved, its greatest fixpoint there exists. It is the least one on ''L'', so ''P'' has least and greatest elements, or more generally that every monotone function on a complete lattice has least and greatest fixpoints.
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| If ''a'' ∈ ''L'' and ''b'' ∈ ''L'', we'll write [''a'', ''b''] for the closed interval with bounds ''a'' and ''b'': { x ∈ ''L'' | ''a'' ≤ x ≤ ''b'' }. If ''a'' ≤ ''b'', then [''a'', ''b''] is a complete lattice.
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| It remains to be proven that P is complete lattice. Let <math>1_L = \bigvee L</math>, ''W'' ⊆ ''P'' and <math>w = \bigvee W</math>. We′ll show that ''f''([''w'', ''1<sub>L</sub>'']) ⊆ [''w'', ''1<sub>L</sub>'']. Indeed for every ''x'' ∈ ''W'' we have ''x'' = ''f(x)'' ≤ ''f(w)''. Since ''w'' is the least upper bound of ''W'', ''w'' ≤ ''f(w)''. Then from ''y'' ∈ [''w'', ''1<sub>L</sub>''] follows that ''w'' ≤ ''f(w)'' ≤ ''f(y)'', giving ''f(y)'' ∈ [''w'', ''1<sub>L</sub>''] or simply ''f''([''w'', ''1<sub>L</sub>'']) ⊆ [''w'', ''1<sub>L</sub>'']. This allow us to look at ''f'' as a function on the complete lattice [''w'', ''1<sub>L</sub>'']. Then it has a least fixpoint there, giving us the least upper bound of ''W''. We′ve shown that an arbitrary subset of ''P'' has a supremum, which turns ''P'' into a complete lattice.
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| == See also ==
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| * [[Kleene fixpoint theorem]]
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| * [[Kantorovitch fixpoint theorem]] (known also as Tarski-Kantorovitch fixpoint principle)
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| * [[Modal μ-calculus]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| *{{cite book | author=Andrzej Granas and [[James Dugundji]] | title=Fixed Point Theory | publisher=[[Springer Science+Business Media|Springer-Verlag]], New York | year=2003 | isbn=0-387-00173-5}}
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| *{{cite book|first=T.|last=Forster|title=Logic, Induction and Sets|isbn=0-521-53361-9}}
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| ===Recent developments===
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| *{{cite journal | author=S. Hayashi | title=Self-similar sets as Tarski's fixed points | journal=Publ. RIMS Kyoto Univ. | year=1985 | volume=21 | pages=1059–1066 | doi=10.2977/prims/1195178796 | issue=5}}
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| *{{cite journal | author=J. Jachymski, L. Gajek, K. Pokarowski | title=The Tarski-Kantorovitch principle and the theory of iterated function systems | journal=Bull. Austral. Math. Soc. | year=2000 | volume=61 | pages=247–261 | doi=10.1017/S0004972700022243 | issue=02}}
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| *{{cite journal | author=E.A. Ok | title=Fixed set theory for closed correspondences with applications to self-similarity and games | journal=Nonlinear Anal. | year=2004 | volume=56 | pages=309–330 | doi=10.1016/j.na.2003.08.001 | issue=3}}
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| *{{cite journal | author=B.S.W. Schröder | title=Algorithms for the fixed point property | journal=Theoret. Comput. Sci. | year=1999 | volume=217 | pages=301–358 | doi=10.1016/S0304-3975(98)00273-4 | issue=2}}
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| *{{cite journal | author=S. Heikkilä | title=On fixed points through a generalized iteration method with applications to differential and integral equations involving discontinuities| journal=Nonlinear Anal. | year=1990 | volume=14 | pages=413–426 | doi=10.1016/0362-546X(90)90082-R | issue=5}}
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| *{{cite journal | author=R. Uhl | title=Smallest and greatest fixed points of quasimonotone increasing mappings | journal=[[Mathematische Nachrichten]] | year=2003 | volume=248–249 | pages=204–210 | doi=10.1002/mana.200310016}}
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| ==External links==
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| * J. B. Nation, [http://bigcheese.math.sc.edu/~mcnulty/alglatvar/ ''Notes on lattice theory''].
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| {{DEFAULTSORT:Knaster-Tarski theorem}}
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| [[Category:Order theory]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Fixed-point theorems]]
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| [[Category:Theorems in the foundations of mathematics]]
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| [[Category:Articles containing proofs]]
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Hello, I'm Cory Robbs. My spouse does not like it just how I do but what I actually like doing is bodybuilding and today I have time for you to accept new issues. For a long time I Have been living in Tennessee. Releasing production is my day job now but I intend on changing it.
Stop by my homepage Jordan Kurland