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In [[abstract algebra]], the idea of an '''inverse element''' generalises concepts of a [[additive inverse|negation (sign reversal)]] in relation to [[addition]], and a [[Multiplicative inverse|reciprocal]] in relation to [[multiplication]]. The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a [[Group (mathematics)|group]].
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The word 'inverse' is derived from {{lang-la|[[wikt:inversus|inversus]]}} that means 'turned upside down', 'overturned'.
 
== Formal definitions ==
=== In a unital magma ===
Let <math>S</math> be a [[Set (mathematics)|set]] with a [[binary operation]] <math>*</math> (i.e., a [[Magma (algebra)|magma]]). If <math>e</math> is an [[identity element]] of <math>(S,*)</math> (i.e., ''S'' is a unital magma) and <math>a*b=e</math>, then <math>a</math> is called a '''left inverse''' of <math>b</math> and <math>b</math> is called a '''right inverse''' of <math>a</math>. If an element <math>x</math> is both a left inverse and a right inverse of <math>y</math>, then <math>x</math> is called a '''two-sided inverse''', or simply an '''inverse''', of <math>y</math>. An element with a two-sided inverse in <math>S</math> is called '''invertible''' in <math>S</math>. An element with an inverse element only on one side is '''left invertible''', resp. '''right invertible'''. If all elements in ''S'' are invertible, ''S'' is called a [[Quasigroup|loop]].
 
Just like <math>(S,*)</math> can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a ''two-sided'' identity <math>e</math>). It can even have several left inverses ''and'' several right inverses.
 
If the operation <math>*</math> is [[associative]] then if an element has both a left inverse and a right inverse, they are equal. In other words, in a [[monoid]] every element has at most one inverse (as defined in this section). In a monoid, the set of (left and right) invertible elements is a [[group (mathematics)|group]], called the [[group of units]] of <math>S</math>, and denoted by <math>U(S)</math> or ''H''<sub>1</sub>.
 
A left-invertible element is left-[[cancellation property|cancellative]], and analogously for right and two-sided.
 
=== In a semigroup ===
{{main|Regular semigroup}}
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity, i.e. in a [[semigroup]].
 
In a semigroup <math>S</math> an element ''x'' is called '''(von Neumann) regular''' if there exists some element ''z'' in S such that ''xzx'' = ''x''; ''z'' is sometimes called a '''pseudoinverse'''. An element ''y'' is called (simply) an '''inverse''' of ''x'' if ''xyx'' = ''x'' and ''y'' = ''yxy''. Every regular element has at least one inverse: if ''x'' = ''xzx'' then it is easy to verify that ''y'' = ''zxz'' is an inverse of ''x'' as defined in this section. Another easy to prove fact: if ''y'' is an inverse of ''x'' then ''e'' = ''xy'' and ''f'' = ''yx'' are [[idempotent element|idempotent]]s, that is ''ee'' = ''e'' and ''ff'' = ''f''. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ''ex'' = ''xf'' = ''x'', ''ye'' = ''fy'' = ''y'', and ''e'' acts as a left identity on ''x'', while ''f'' acts a right identity, and the left/right roles are reversed for ''y''. This simple observation can be generalized using [[Green's relations]]: every idempotent ''e'' in an arbitrary semigroup is a left identity for ''R<sub>e</sub>'' and right identity for ''L<sub>e</sub>''.<ref>Howie, prop. 2.3.3, p. 51</ref> An intuitive description of this is fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.
 
In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in [[Green's_relations#The_H_and_D_relations|''H''<sub>1</sub>]] have an inverse from the unital magma perspective, whereas for any idempotent ''e'', the elements of ''H''<sub>e</sub> have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an [[inverse semigroup]]. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an [[absorbing element]] 0 because 000 = 0, whereas a group may not.
 
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a '''quasi-inverse'''. This is generally justified because in most applications (e.g. all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity.
 
=== ''U''-semigroups ===
 
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (''a''°)° = ''a'' for all ''a'' in ''S''; this endows ''S'' with a type {{langle}}2,1{{rangle}} algebra. A semigroup endowed with such an operation is called a '''''U''-semigroup'''. Although it may seem that ''a''° will be the inverse of ''a'', this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of ''U''-semigroups have been studied:
 
* '''''I''-semigroups''', in which the interaction axiom is ''aa''°''a'' = ''a''
* '''[[Semigroup with involution|*-semigroups]]''', in which the interaction axiom is (''ab'')° = ''b''°''a''°. Such an operation is called an [[involution (mathematics)|involution]], and typically denoted by ''a''*
 
Clearly a group is both an ''I''-semigroup and a *-semigroup. Inverse semigroups are exactly those semigroups that are both ''I''-semigroups and *-semigroups. A class of semigroups important in semigroup theory are [[completely regular semigroup]]s; these are ''I''-semigroups in which one additionally has ''aa''° = ''a''°''a''; in other words every element has commuting pseudoinverse ''a''°. There are few concrete examples of such semigroups however; most are [[completely simple semigroup]]s. In contrast, a class of *-semigroups, the [[*-regular semigroup]]s, yield one of best known examples of a (unique) pseudoinverse, the [[Moore–Penrose inverse]]. In this case however the involution ''a''* is not the pseudoinverse. Rather, the pseudoinverse of ''x'' is the unique element ''y'' such that ''xyx'' = ''x'', ''yxy'' = ''y'',  (''xy'')* = ''xy'', (''yx'')* = ''yx''. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the '''generalized inverse''' or '''Penrose–Moore inverse'''. In a *-regular semigroup ''S'' one can identify a special subset of idempotents F(''S'') called a [[Semigroup with involution#P-systems|P-system]]; every element ''a'' of the semigroup has exactly one inverse ''a''* such that ''aa''* and ''a''*''a'' are in F(''S''). The P-systems of Yamada are based upon the notion of regular *-semigroup as defined by Nordahl and Scheiblich.  
 
== Examples ==
All examples in this section involve associative operators, thus we shall use the terms left/right inverse for the unital magma-based definition, and quasi-inverse for its more general version.
 
=== Real numbers ===
Every [[real number]] <math>x</math> has an [[additive inverse]] (i.e. an inverse with respect to [[addition]]) given by <math>-x</math>. Every nonzero real number <math>x</math> has a [[multiplicative inverse]] (i.e. an inverse with respect to [[multiplication]]) given by <math>\frac 1{x}</math> (or <math>x^{-1}</math>). By contrast, [[0 (number)|zero]] has no multiplicative inverse, but it has a unique quasi-inverse, 0 itself.
 
=== Functions and partial functions ===
A function <math>g</math> is the left (resp. right) [[inverse function|inverse of a function]] <math>f</math> (for [[function composition]]), if and only if <math>g \circ f</math> (resp. <math>f \circ g</math>) is the [[identity function]] on the [[domain of a function|domain]] (resp. [[codomain]]) of <math>f</math>. The inverse of a function <math>f</math> is often written <math>f^{-1}</math>, but [[inverse function#Note on notation|this notation is sometimes ambiguous]]. Only [[bijection]]s have two-sided inverses, but ''any'' function has a quasi-inverse, i.e. the [[full transformation monoid]] is regular. The monoid of [[partial functions]] is also regular, whereas the [[symmetric inverse semigroup|monoid of injective partial transformations]] is the prototypical inverse semigroup.
 
=== Galois connections ===
The lower and upper adjoints in a (monotone) [[Galois connection]], ''L'' and ''G'' are quasi-inverses of each other, i.e. ''LGL'' = ''L'' and ''GLG'' = ''G'' and one uniquely determines the other. They are not left or right inverses of each other however.
 
=== Matrices ===
A [[square matrix]] <math>M</math> with entries in a [[field (mathematics)|field]] <math>K</math> is invertible (in the set of all square matrices of the same size, under [[matrix multiplication]]) if and only if its [[determinant]] is different from zero. If the determinant of <math>M</math> is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See [[invertible matrix]] for more.
 
More generally, a square matrix over a [[commutative ring]] <math>R</math> is invertible [[if and only if]] its determinant is invertible in <math>R</math>.
 
Non-square matrices of [[full rank]] have several one-sided inverses:<ref>[http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture33.htm MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse.]</ref>
* For <math>A:m\times n \mid m>n</math> we have a left inverse: <math> \underbrace{ (A^{T}A)^{-1}A^{T} }_{ A^{-1}_\text{left} } A = I_{n} </math>
* For <math>A:m\times n \mid m<n</math> we have a right inverse: <math> A \underbrace{ A^{T}(AA^{T})^{-1} }_{ A^{-1}_\text{right} } = I_{m} </math>
 
The right inverse can be used to determine the least norm solution of ''Ax'' = ''b''.
 
No rank-deficient matrix has any (even one-sided) inverse.  However, the [[Moore–Penrose pseudoinverse]] exists for all matrices, and coincides with the left or right (or true) inverse when it exists.
 
As an example of matrix inverses, consider:
 
: <math>A:2\times 3 =
  \begin{bmatrix}
    1 & 2 & 3 \\
    4 & 5 & 6
  \end{bmatrix}
</math>
 
So, as ''m'' < ''n'', we have a right inverse. <math>A^{-1}_\text{right} = A^{T}(AA^{T})^{-1}</math>
 
: <math>\begin{align}
  AA^{T} &= \begin{bmatrix}
              1 & 2 & 3 \\
              4 & 5 & 6
            \end{bmatrix}\cdot
            \begin{bmatrix}
              1 & 4\\
              2 & 5\\
              3 & 6
            \end{bmatrix} = 
            \begin{bmatrix}
              14 & 32\\
              32 & 77
            \end{bmatrix} \\
  (AA^{T})^{-1} &= \begin{bmatrix}
                    14 & 32\\
                    32 & 77
                  \end{bmatrix}^{-1} = \frac{1}{54}
                  \begin{bmatrix}
                      77 & -32\\
                    -32 & 14
                  \end{bmatrix} \\
  A^{T}(AA^{T})^{-1} &= \frac{1}{54}
                  \begin{bmatrix}
                    1 & 4\\
                    2 & 5\\
                    3 & 6
                  \end{bmatrix}\cdot
                  \begin{bmatrix}
                      77 & -32\\
                    -32 & 14
                  \end{bmatrix} = \frac{1}{18}
                  \begin{bmatrix}
                    -17 & 8\\
                      -2 & 2\\
                      13 & -4
                  \end{bmatrix} = A^{-1}_\text{right}
\end{align}</math>
 
The left inverse doesn't exist, because
 
: <math>
  A^{T}A = \begin{bmatrix}
            1 & 4\\
            2 & 5\\
            3 & 6
          \end{bmatrix} \cdot
          \begin{bmatrix}
            1 & 2 & 3 \\
            4 & 5 & 6
          \end{bmatrix} =
          \begin{bmatrix}
            17 & 22 & 27 \\
            22 & 29 & 36\\
            27 & 36 & 45
          \end{bmatrix}
</math>
 
which is a singular matrix, and cannot be inverted.
 
== See also ==
*[[loop (algebra)]]
*[[division ring]]
*[[unit (ring theory)]]
*[[Latin square property]]
 
==Notes==
{{Reflist}}
 
==References==
* M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 15 (def in unital magma) and p. 33 (def in semigroup)
*{{cite book|last= Howie|first= John M.|title=Fundamentals of Semigroup Theory|year=1995|publisher=[[Clarendon Press]]|isbn=0-19-851194-9}} contains all of the semigroup material herein except *-regular semigroups.
* Drazin, M.P., ''Regular semigroups with involution'', Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
* Miyuki Yamada, ''P-systems in regular semigroups'', Semigroup Forum, 24(1), December 1982, pp. 173–187
*Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.
 
[[Category:Algebra]]
[[Category:Abstract algebra]]
[[Category:Binary operations|*Inverse element]]

Revision as of 10:42, 1 March 2014

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Eat at an even more relaxing tempo. Lots of people are usually extremely occupied and as a result, get into the habit of smoking of eating rapidly. Take some time biting the food, and don't gulp the food lower. Loosen up when you try to eat and little by little consume each mouthful of food. If you slow down, you may really feel happy and then eat less. It will likewise stop you from overindulging.

Drink plenty of water rather than other cocktails whenever possible. Offer juices or milk on 1 or 2 food and try not to offer it continuously. Constant use of milk or liquid throughout the day will spoil their appetite.

If you are planning out, share a plate together with your time or with a good friend. You could make the best food choice possible yet still wind up having a huge providing, packed with unhealthy calories. Discussing meals in the diner can help you decrease your servings and save money. This will enable you to step out to enjoy while maintaining a healthy diet plan.

For a snack food that may be healthful and tastes fantastic, try out pureeing berries, pears or peaches. This fairly sweet spread out is fantastic for dipping pretzels or pitas. Combine your fresh fruits choices so you don't become bored of the same fruits over and over again.

Something that could be microwaved should be prevented. These foods normally have a lot of additives that are very best avoided.

Consider new and trendy tasty recipes if you are planning on engaging in a whole new diet regime. You possibly can make you have shakes, healthy protein pubs and do-it-yourself peanut butter. These are super easy to make and useful to you. Another wholesome choice is oatmeal pancakes.

Ginger herb is a marvellous, natural way to ease movements disease. You can get ginger supplements. Commencing around an hour or so ahead of the suggestion, you must eat approximately 1,000 milligrams than it. Repeat this approximately every three hrs. Ginger is great for preventing the tummy cramps and nausea typically related with vacationing. Ginger herb sweets and herbal tea can even be beneficial.

In the event you goal would be to lessen the amount of glucose in your daily diet, be watchful of those food products which have corn syrup, as which is truly glucose. There are lots of foods and condiments including corn syrup so make sure you spend some time to look at the element lists on meals brands.

Transform out white colored a loaf of bread for total-grain seeded loaves of bread on your own sandwich. Wholegrain loaves of bread are reduced glycemic, therefore they help to quit the food cravings and job to keep your weight in balance. Also, whole grain loaves of bread has many dietary fiber and healthy body fat.

Very good nourishment is so essential to our well-being and health and when you eat a wholesome, balanced diet program, you will look and feel your best. Among the finest approaches to raise the healthy top quality of the diet is by reducing the volume of highly processed glucose that you try to eat. Monitor your usage of soft drink as well as other sweet cocktails. You are hoping to avoid sugar and they cocktails have a lot of it. You may lose fat by merely cutting your sweets consumption. Moreover, you will possess much more electricity and appear better.

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Ensure that you eat ample food items. It is actually tempting for individuals to constrain their diet plan drastically when trying to lose weight. If you do not obtain the suitable nutrients, you could see a decrease in both your insulin and blood sugar. Try to eat 5 or half a dozen little dishes each day to get the most nutrition from the food items.

Fresh beets are ideal for you, but canned beets aren't. New beets have a lot of dietary fiber plus some excellent nutrients, but processed beets often times have a lot of additional salt. Vapor the plants and put the origins to the salads to utilize the complete plant.

A good nutrition hint for anyone is usually to consume a balanced diet abundant in the appropriate amounts of wholesome fats, protein and carbohydrates. Essentially, you diet program should include of about 30% healthy proteins, 50% carbohydrates and 20Percent or a smaller amount of body fat. Virtually all your dietary carbs needs to be one thing aside from easy sugars.

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If you have kids, take time to make them learn about diet. It is essential to supply them healthy food now. This will assist them form wholesome ways of eating down the road. Reduce the sweets and starchy foods they eat, as this can be unfavorable towards the system.

Are you currently now better educated about nutrients? Do you experience feeling like you will have a much better concept of the way to devise an effective nutrients prepare? Are you feeling you can actually harmony food products you need to be ingesting with food products you wish to take in? Is your program working for you? With any luck ,, the info and methods with this report gave the right strategies to the above mentioned inquiries.

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