5493:
4930:
5488:{\displaystyle {\begin{aligned}AA^{\text{T}}&={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}={\begin{bmatrix}14&32\\32&77\end{bmatrix}}\\\left(AA^{\text{T}}\right)^{-1}&={\begin{bmatrix}14&32\\32&77\end{bmatrix}}^{-1}={\frac {1}{54}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}\\A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}&={\frac {1}{54}}{\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}77&-32\\-32&14\end{bmatrix}}={\frac {1}{18}}{\begin{bmatrix}-17&8\\-2&2\\13&-4\end{bmatrix}}=A_{\text{right}}^{-1}\end{aligned}}}
5692:
3903:
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
2441:
A matrix has a left inverse if and only if its rank equals its number of columns. This left inverse is not unique except for square matrices where the left inverse equal the inverse matrix. Similarly, a right inverse exists if and only if the rank equals the number of rows; it is not unique in the
5504:
3680:
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; that is, in a
4622:
4469:
2801:, respectively. An homomorphism of algebraic structures that has a left inverse or a right inverse is respectively injective or surjective, but the converse is not true in some algebraic structures. For example, the converse is true for
4827:
2414:
is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings.
5687:{\displaystyle A^{\text{T}}A={\begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}}{\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}={\begin{bmatrix}17&22&27\\22&29&36\\27&36&45\end{bmatrix}}}
4922:
1854:
Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the
3919:. This is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see
4935:
1062:
4734:
2782:. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an
2757:
631:
4515:
4362:
2557:
1276:
2648:
4520:
4367:
489:
308:
279:
4742:
matrix has any (even one-sided) inverse. However, the Moore–Penrose inverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.
2517:
1135:
1603:
2175:
1459:
1390:
2213:(that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the
1490:
1097:
958:
3883:
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 the Green class
2614:
2940:
2583:
1426:
749:
2145:
1356:
3969:, 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
2908:
1701:
864:
3435:
3337:
3235:
2972:
1954:
1893:
4654:
3657:
1922:
1018:
708:
3880:. An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.
1981:
1557:
1531:
1319:
786:
679:
1767:
1744:
989:
413:
4307:
4283:
4253:
4225:
4202:
3628:
3592:
3192:
3168:
3148:
3120:
3100:
3080:
3060:
3036:
3016:
2992:
2879:
2848:
2704:
2672:
1721:
549:
4751:
1775:
represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order.
2422:(that is, matrices with integer entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a
897:
Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on
4843:
4255:
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
3602:(an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a
1828:, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).
924:
An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the
512:
for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.
4110:. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the
964:, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps
4667:
1023:
1392:
When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in
1186:
In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if
200:, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).
5788:
2083:. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.
2723:
582:
2237:
2014:
5834:
5814:
2218:
2228:(that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not
3884:
252:, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of
164:
has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the
4476:
4323:
4059:
4031:
3934:
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (
2809:
over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a
1670:
1168:
1204:
4617:{\displaystyle A\underbrace {A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}} _{A_{\text{right}}^{-1}}=I_{m}}
4055:
3237:
may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
20:
4464:{\displaystyle \underbrace {\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}} _{A_{\text{left}}^{-1}}A=I_{n}}
5889:
3444:
Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table
3908:. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an
2522:
5884:
4063:
3998:
3598:
then if an element has both a left inverse and a right inverse, they are equal. In other words, in a
1164:
432:
284:
255:
215:. They are also commonly used for operations that are not defined for all possible operands, such as
2490:
4015:
2768:
1659:
1562:
1102:
5789:"MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse"
4173:
and one uniquely determines the other. They are not left or right inverses of each other however.
2619:
1431:
2150:
2091:
1365:
934:
60:
3954:
with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called a
1468:
1075:
2470:
2107:
1832:
1651:
1493:
925:
366:
249:
177:
46:
2913:
2562:
1395:
713:
5757:
4228:
2296:
2264:
2252:. Localization is also used with zero divisors, but, in this case the original ring is not a
2117:
1856:
1788:
1772:
1644:
1328:
792:
516:
354:
4062:(in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the
3915:
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a
2887:
2442:
case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.
1676:
834:
5749:
4127:
3859:
3408:
3310:
3208:
2945:
2806:
2707:
2675:
2450:
2268:
1927:
1866:
1836:
1813:
is a monoid for ring multiplication. In this case, the invertible elements are also called
1462:
1145:
961:
820:
520:
358:
346:
345:
that are everywhere defined (that is, the operation is defined for any two elements of its
4630:
3633:
2588:
1898:
994:
684:
8:
4661:
4205:
3920:
3722:
3603:
2462:
2272:
2210:
2199:
2034:
1960:
1848:
1803:
1636:
1536:
1510:
1298:
898:
765:
658:
248:
that means 'turned upside down', 'overturned'. This may take its origin from the case of
204:
1749:
1726:
971:
395:
5725:
4292:
4268:
4238:
4210:
4187:
3799:
3613:
3577:
3177:
3153:
3133:
3105:
3085:
3065:
3045:
3021:
3001:
2977:
2864:
2833:
2814:
2689:
2657:
2388:
2383:
is an invertible element under matrix multiplication. A matrix over a commutative ring
2276:
2245:
2103:
2030:
2006:
1986:
1844:
1840:
1825:
1814:
1810:
1706:
823:, and the composition of the identity functions of two different sets are not defined.
534:
212:
208:
197:
189:
181:
4822:{\displaystyle A:2\times 3={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}}
4030:-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are
5830:
5810:
5753:
5745:
4256:
4146:
3909:
3905:
3818:. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and
3675:
2855:
2810:
2772:
2764:
2454:
2423:
2380:
2061:
1640:
906:
816:
552:
385:
377:
must be extended to partial operations; this is the object of the first subsections.
350:
342:
5879:
5744:
The usual definition of an identity element has been generalized for including the
4263:
2882:
2859:
2783:
2419:
2404:
2290:
2225:
2073:
2018:
1792:
1288:
914:
564:
374:
318:
220:
216:
173:
68:
35:
4917:{\displaystyle A_{\text{right}}^{-1}=A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}.}
392:) on which a partial operation (possibly total) is defined, which is denoted with
5863:
5856:
5826:
5698:
2851:
2357:
2249:
2241:
2214:
2010:
1998:
1616:
1160:
805:
224:
3904:
exactly one inverse as defined in this section, then the semigroup is called an
2256:
of the localisation; instead, it is mapped non-injectively to the localization.
5720:
5715:
4739:
4286:
3607:
3304:
2400:
2090:; this means that multiplication is associative and has an identity called the
1994:
1818:
1065:
918:
239:
3892:
have an inverse from the unital magma perspective, whereas for any idempotent
1723:
defines a transformation that is the inverse of the transformation defined by
5873:
5710:
4657:
4182:
2365:
2361:
2280:
2229:
2203:
2049:
812:, and two identity matrices of different size cannot be multiplied together.
370:
5807:
Monoids, Acts and
Categories with Applications to Wreath Products and Graphs
244:
3438:
3340:
2802:
2458:
1666:
1647:
that has an identity element, and for which every element has an inverse.
1608:
754:
It follows that a total operation has at most one identity element, and if
389:
5841:
contains all of the semigroup material herein except *-regular semigroups.
5809:, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000,
2430:. A square integer matrix is unimodular if and only if its determinant is
353:, that is operations that are not defined everywhere. Common examples are
4232:
4227:
is invertible (in the set of all square matrices of the same size, under
3595:
2481:
2427:
2300:
2057:
2053:
1624:
1612:
1359:
423:
314:
228:
157:
27:
4054:°. There are few concrete examples of such semigroups however; most are
1989:
construction. This is the method that is commonly used for constructing
1746:
that is, the transformation that "undoes" the transformation defined by
1662:, since the inverse of the inverse of an element is the element itself.
1654:
from the group to itself that may also be considered as an operation of
2798:
1153:
172:. Often an adjective is added for specifying the operation, such as in
2438:, since these two numbers are the only units in the ring of integers.
4627:
The left inverse can be used to determine the least norm solution of
4313:
3682:
3571:
the only element with a two-sided inverse is the identity element 1.
2794:
2779:
2184:
1149:
5817:, p. 15 (def in unital magma) and p. 33 (def in semigroup)
2793:
A function has a left inverse or a right inverse if and only it is
2683:
2651:
2466:
2284:
1620:
362:
232:
3405:
every element has a unique two-sided inverse (namely itself), but
3303:
A unital magma in which all elements are invertible need not be a
3854:
acts a right identity, and the left/right roles are reversed for
2426:
for distinguishing it from matrices that are invertible over the
2253:
2002:
1990:
1183:
under an operation if it has a left inverse and a right inverse.
917:. This lack of inverses is the main motivation for extending the
1057:{\textstyle n\mapsto \left\lfloor {\frac {n}{2}}\right\rfloor ,}
4729:{\displaystyle x=\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}b.}
3599:
2817:. This terminology is also used for morphisms in any category.
2087:
1784:
2360:, that is, an identity element for matrix multiplication is a
1284:
of an invertible element is its unique left or right inverse.
1655:
644:
for which the left-hand sides of the equalities are defined.
4137:
All examples in this section involve associative operators.
2650:
In the function and homomorphism cases, this means that the
1287:
If the operation is denoted as an addition, the inverse, or
4050:; in other words every element has commuting pseudoinverse
3774:
as defined in this section. Another easy to prove fact: if
5862:
Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups,
2110:(for avoiding confusion with additive inverses) of a unit
5848:, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
4070:* is not the pseudoinverse. Rather, the pseudoinverse of
3194:. An element with an inverse element only on one side is
2445:
2232:(that is, their product with a nonzero element cannot be
2045:, which are denoted as the usual operations on numbers.
2477:, and share many properties with function composition.
2364:(same number for rows and columns) whose entries of the
3300:
the elements 2 and 3 each have two two-sided inverses.
1194:
are respectively a left inverse and a right inverse of
349:). However, these concepts are also commonly used with
5626:
5577:
5529:
5408:
5353:
5305:
5199:
5141:
5060:
5009:
4963:
4778:
2153:
1471:
1368:
1105:
1078:
1026:
289:
260:
16:
Generalization of additive and multiplicative inverses
5507:
4933:
4846:
4754:
4670:
4633:
4523:
4479:
4370:
4326:
4295:
4271:
4241:
4213:
4190:
3754:. Every regular element has at least one inverse: if
3636:
3616:
3580:
3411:
3313:
3211:
3180:
3156:
3136:
3108:
3088:
3068:
3048:
3024:
3004:
2980:
2948:
2916:
2890:
2867:
2836:
2726:
2692:
2660:
2622:
2591:
2565:
2525:
2493:
2120:
1963:
1930:
1901:
1869:
1752:
1729:
1709:
1679:
1565:
1539:
1513:
1434:
1398:
1331:
1301:
1207:
997:
974:
937:
837:
768:
716:
687:
661:
585:
537:
435:
398:
287:
258:
2752:{\displaystyle \operatorname {id} _{X}\colon X\to X}
2244:
from the ring of integers, and, more generally, the
4176:
3858:. This simple observation can be generalized using
328:
5686:
5487:
4916:
4821:
4728:
4648:
4616:
4509:
4463:
4356:
4301:
4277:
4247:
4219:
4196:
3651:
3622:
3586:
3429:
3331:
3229:
3186:
3162:
3142:
3114:
3094:
3074:
3054:
3030:
3010:
2986:
2966:
2934:
2902:
2873:
2842:
2751:
2698:
2666:
2642:
2608:
2577:
2551:
2511:
2387:is invertible if and only if its determinant is a
2275:, and straightforwardly extended to matrices over
2169:
2139:
1975:
1948:
1916:
1887:
1839:. In this monoid, the invertible elements are the
1761:
1738:
1715:
1695:
1597:
1551:
1525:
1484:
1453:
1420:
1384:
1350:
1313:
1270:
1152:, and it has a right inverse if and only if it is
1144:More generally, a function has a left inverse for
1129:
1091:
1056:
1012:
983:
952:
858:
780:
743:
710:(This results immediately from the definition, by
702:
673:
626:{\displaystyle x*e=x\quad {\text{and}}\quad e*y=y}
625:
543:
483:
407:
302:
273:
3866:in an arbitrary semigroup is a left identity for
515:Examples of non-total associative operations are
19:"Invertible" redirects here. For other uses, see
5871:
3437:is not a loop because the Cayley table is not a
2778:A function is invertible if and only if it is a
4058:. In contrast, a subclass of *-semigroups, the
1843:; the elements that have left inverses are the
113:. (An identity element is an element such that
4235:is different from zero. If the determinant of
3082:is both a left inverse and a right inverse of
2240:, which produces, in particular, the field of
321:, except when otherwise stated and in section
4145:The lower and upper adjoints in a (monotone)
2183:is never a unit, except when the ring is the
1847:, and those that have right inverses are the
4745:As an example of matrix inverses, consider:
3912:0 because 000 = 0, whereas a group may not.
2682:. In the morphism case, this means that the
2616:or, in the function and homomorphism cases,
2147:or, when the multiplication is commutative,
369:. It follows that the common definitions of
184:. In this case (associative operation), an
149:for which the left-hand sides are defined.)
4157:are quasi-inverses of each other; that is,
2202:if the multiplication is commutative, or a
211:—where invertible elements are also called
4038:-semigroups in which one additionally has
826:
3150:. An element with a two-sided inverse in
1985:This extension of a monoid is allowed by
5859:, 24(1), December 1982, pp. 173–187
5498:The left inverse doesn't exist, because
960:has infinitely many left inverses under
207:—where every element is invertible, and
188:is an element that has an inverse. In a
4262:More generally, a square matrix over a
2295:, because of the use of the concept of
1835:from a set to itself is a monoid under
1673:of this set. In this case, the inverse
928:from the integers to the integers. The
5872:
4510:{\displaystyle A:m\times n\mid m<n}
4357:{\displaystyle A:m\times n\mid m>n}
4066:. In this case however the involution
2446:Functions, homomorphisms and morphisms
2289:in this section, only matrices over a
870:is an identity element, one says that
5820:
4517:we have right inverses; for example,
4140:
4002:, in which the interaction axiom is (
2473:into operations that are also called
4364:we have left inverses; for example,
3983:, in which the interaction axiom is
2825:
2198:is the only non-unit, the ring is a
1271:{\displaystyle l=l*(x*r)=(l*x)*r=r.}
655:are two identity elements such that
526:
322:
313:In this article, the operations are
5805:M. Kilp, U. Knauer, A.V. Mikhalev,
2786:. In the other cases, one talks of
1020:is a right inverse of the function
238:The word 'inverse' is derived from
13:
5846:Regular semigroups with involution
4014:°. Such an operation is called an
2820:
2086:Under multiplication, a ring is a
1559:is invertible, and its inverse is
14:
5901:
4316:have several one-sided inverses:
4289:its determinant is invertible in
4177:Generalized inverses of matrices
3669:
2219:infinite-dimensional vector space
2060:; it has an identity, called the
1863:has the cancellation property if
1163:, right inverses are also called
5823:Fundamentals of Semigroup Theory
4924:By components it is computed as
2552:{\displaystyle g\colon Y'\to Z,}
2480:In all the case, composition is
913:is the only element that has an
417:
329:Definitions and basic properties
5853:P-systems in regular semigroups
3973:-semigroups have been studied:
3926:
3762:then it is easy to verify that
2320:matrix (that is, a matrix with
2102:for multiplication is called a
2052:, which means that addition is
1167:, and left inverses are called
762:are different identities, then
607:
601:
555:associative operation on a set
484:{\displaystyle x*(y*z)=(x*y)*z}
303:{\displaystyle {\tfrac {y}{x}}}
274:{\displaystyle {\tfrac {x}{y}}}
223:. This has been generalized to
5781:
5772:
5763:
5738:
3646:
3640:
3424:
3412:
3326:
3314:
3224:
3212:
2743:
2540:
2512:{\displaystyle f\colon X\to Y}
2503:
1250:
1238:
1232:
1220:
1130:{\textstyle {\frac {n-1}{2}},}
1030:
941:
472:
460:
454:
442:
203:Inverses are commonly used in
1:
5799:
4032:completely regular semigroups
3700:if there exists some element
2674:equals or is included in the
2048:Under addition, a ring is an
1778:
1598:{\displaystyle y^{-1}x^{-1}.}
4121:
4056:completely simple semigroups
3961:. Although it may seem that
3307:. For example, in the magma
2643:{\displaystyle Y\subset Y'.}
2372:, and all other entries are
2356:, and only in this case. An
2170:{\textstyle {\frac {1}{x}}.}
1831:For example, the set of the
1630:
1454:{\displaystyle f^{\circ -1}}
1385:{\textstyle {\frac {1}{x}}.}
791:For example, in the case of
34:generalises the concepts of
7:
5704:
4840:, we have a right inverse,
4132:
4026:Clearly a group is both an
4018:, and typically denoted by
3846:acts as a left identity on
3205:Elements of a unital magma
2771:), which is called also an
2395:(that is, is invertible in
2259:
2072:has an inverse, called its
2024:
1485:{\textstyle {\frac {1}{f}}}
1174:
1092:{\textstyle {\frac {n}{2}}}
953:{\displaystyle x\mapsto 2x}
808:for every positive integer
227:, where, by definition, an
21:Invertible (disambiguation)
10:
5906:
5701:, and cannot be inverted.
4125:
3673:
2585:is defined if and only if
2236:). This is the process of
1321:Otherwise, the inverse of
819:are identity elements for
517:multiplication of matrices
312:
18:
5769:Howie, prop. 2.3.3, p. 51
5756:as identity elements for
5748:as identity elements for
3965:° will be the inverse of
2767:, algebraic structure or
2005:and, more generally, the
1461:is not commonly used for
341:are commonly defined for
5731:
3464:
3461:
3458:
3455:
3452:
3449:
3356:
3353:
3350:
3347:
3251:
3248:
3245:
3242:
2935:{\displaystyle a,b\in S}
2578:{\displaystyle g\circ f}
2267:is commonly defined for
1806:under monoid operation.
1421:{\displaystyle x^{*-1}.}
744:{\displaystyle e=e*f=f.}
5821:Howie, John M. (1995).
4312:Non-square matrices of
3873:and right identity for
2191:as its unique element.
2140:{\displaystyle x^{-1},}
2092:multiplicative identity
1650:Thus, the inverse is a
1351:{\displaystyle x^{-1},}
827:Left and right inverses
519:of arbitrary size, and
422:A partial operation is
5688:
5489:
4918:
4823:
4730:
4650:
4618:
4511:
4465:
4358:
4303:
4279:
4249:
4221:
4198:
4074:is the unique element
3730:is called (simply) an
3720:is sometimes called a
3653:
3624:
3588:
3431:
3333:
3231:
3188:
3164:
3144:
3116:
3096:
3076:
3056:
3032:
3012:
2988:
2968:
2936:
2904:
2903:{\displaystyle e\in S}
2875:
2844:
2775:in the function case.
2753:
2700:
2668:
2644:
2610:
2579:
2553:
2513:
2179:The additive identity
2171:
2141:
2108:multiplicative inverse
1977:
1950:
1918:
1889:
1763:
1740:
1717:
1697:
1696:{\displaystyle g^{-1}}
1599:
1553:
1527:
1494:multiplicative inverse
1486:
1455:
1422:
1386:
1352:
1315:
1272:
1131:
1093:
1058:
1014:
985:
954:
860:
859:{\displaystyle x*y=e,}
782:
745:
704:
675:
627:
545:
485:
409:
323:§ Generalizations
304:
275:
243:
178:multiplicative inverse
5758:matrix multiplication
5716:Latin square property
5689:
5490:
4919:
4824:
4731:
4651:
4619:
4512:
4466:
4359:
4304:
4280:
4250:
4231:) if and only if its
4229:matrix multiplication
4222:
4199:
4116:Moore–Penrose inverse
4064:Moore–Penrose inverse
3698:(von Neumann) regular
3654:
3625:
3589:
3432:
3430:{\displaystyle (S,*)}
3334:
3332:{\displaystyle (S,*)}
3232:
3230:{\displaystyle (S,*)}
3189:
3165:
3145:
3117:
3097:
3077:
3057:
3033:
3013:
2989:
2969:
2967:{\displaystyle a*b=e}
2937:
2905:
2876:
2845:
2754:
2701:
2669:
2645:
2611:
2580:
2554:
2514:
2403:can be computed with
2265:Matrix multiplication
2172:
2142:
2037:with two operations,
1978:
1951:
1949:{\displaystyle yx=zx}
1919:
1890:
1888:{\displaystyle xy=xz}
1857:cancellation property
1789:associative operation
1764:
1741:
1718:
1698:
1645:associative operation
1600:
1554:
1528:
1487:
1456:
1423:
1387:
1358:or, in the case of a
1353:
1325:is generally denoted
1316:
1273:
1148:if and only if it is
1132:
1094:
1059:
1015:
986:
955:
861:
793:matrix multiplication
783:
746:
705:
676:
628:
546:
486:
410:
355:matrix multiplication
305:
276:
5866:, 16(1978), 369–377.
5750:function composition
5505:
4931:
4844:
4752:
4668:
4656:, which is also the
4649:{\displaystyle Ax=b}
4631:
4521:
4477:
4368:
4324:
4293:
4269:
4239:
4211:
4188:
4128:Quasiregular element
4060:*-regular semigroups
3652:{\displaystyle U(S)}
3634:
3614:
3578:
3409:
3311:
3209:
3178:
3154:
3134:
3106:
3086:
3066:
3046:
3022:
3002:
2978:
2946:
2914:
2888:
2865:
2834:
2788:inverse isomorphisms
2724:
2690:
2658:
2620:
2609:{\displaystyle Y'=Y}
2589:
2563:
2523:
2491:
2463:algebraic structures
2457:that generalizes to
2399:. In this case, its
2342:matrix, the product
2151:
2118:
2068:; and every element
1961:
1928:
1917:{\displaystyle y=z,}
1899:
1867:
1849:surjective functions
1837:function composition
1750:
1727:
1707:
1677:
1563:
1537:
1511:
1507:are invertible, and
1492:can be used for the
1469:
1463:function composition
1432:
1396:
1366:
1329:
1299:
1205:
1146:function composition
1103:
1076:
1024:
1013:{\displaystyle 2n+1}
995:
972:
962:function composition
935:
899:nonnegative integers
835:
821:function composition
766:
714:
703:{\displaystyle e=f.}
685:
659:
583:
535:
521:function composition
433:
396:
359:function composition
285:
256:
30:, the concept of an
5480:
4864:
4598:
4442:
4112:generalized inverse
3921:Generalized inverse
3862:: every idempotent
2211:noncommutative ring
2100:invertible element
2035:algebraic structure
1976:{\displaystyle y=z}
1845:injective functions
1841:bijective functions
1824:If a monoid is not
1802:in a monoid form a
1800:invertible elements
1703:of a group element
1658:one. It is also an
1552:{\displaystyle x*y}
1526:{\displaystyle x*y}
1314:{\displaystyle -x.}
921:into the integers.
781:{\displaystyle e*f}
674:{\displaystyle e*f}
361:and composition of
152:When the operation
5746:identity functions
5726:Unit (ring theory)
5684:
5678:
5612:
5566:
5485:
5483:
5463:
5454:
5384:
5342:
5230:
5166:
5085:
5046:
4998:
4914:
4847:
4819:
4813:
4726:
4646:
4614:
4600:
4581:
4578:
4507:
4461:
4444:
4425:
4422:
4354:
4299:
4275:
4245:
4217:
4204:with entries in a
4194:
4141:Galois connections
3896:, the elements of
3649:
3620:
3584:
3427:
3329:
3227:
3184:
3160:
3140:
3112:
3092:
3072:
3052:
3028:
3008:
2984:
2964:
2932:
2900:
2871:
2840:
2815:split monomorphism
2749:
2696:
2664:
2640:
2606:
2575:
2549:
2509:
2246:field of fractions
2167:
2137:
2007:field of fractions
1987:Grothendieck group
1973:
1946:
1914:
1885:
1773:Rubik's cube group
1762:{\displaystyle g.}
1759:
1739:{\displaystyle g,}
1736:
1713:
1693:
1623:is also called an
1595:
1549:
1523:
1482:
1451:
1418:
1382:
1348:
1311:
1268:
1137:depending whether
1127:
1089:
1054:
1010:
984:{\displaystyle 2n}
981:
950:
856:
817:identity functions
778:
741:
700:
671:
623:
541:
481:
408:{\displaystyle *.}
405:
351:partial operations
339:invertible element
300:
298:
271:
269:
194:invertible element
186:invertible element
182:functional inverse
5890:Binary operations
5754:identity matrices
5515:
5470:
5401:
5298:
5266:
5247:
5192:
5111:
4948:
4894:
4875:
4854:
4717:
4690:
4588:
4557:
4538:
4529:
4527:
4432:
4415:
4388:
4373:
4371:
4302:{\displaystyle R}
4278:{\displaystyle R}
4257:invertible matrix
4248:{\displaystyle M}
4220:{\displaystyle K}
4197:{\displaystyle M}
4147:Galois connection
3910:absorbing element
3906:inverse semigroup
3860:Green's relations
3778:is an inverse of
3770:is an inverse of
3676:Regular semigroup
3630:, and denoted by
3623:{\displaystyle S}
3587:{\displaystyle *}
3574:If the operation
3569:
3568:
3403:
3402:
3298:
3297:
3187:{\displaystyle S}
3163:{\displaystyle S}
3143:{\displaystyle y}
3124:two-sided inverse
3115:{\displaystyle x}
3095:{\displaystyle y}
3075:{\displaystyle x}
3055:{\displaystyle a}
3031:{\displaystyle b}
3011:{\displaystyle b}
2987:{\displaystyle a}
2874:{\displaystyle *}
2843:{\displaystyle S}
2826:In a unital magma
2811:split epimorphism
2773:identity function
2759:for every object
2699:{\displaystyle f}
2667:{\displaystyle f}
2455:partial operation
2424:unimodular matrix
2381:invertible matrix
2368:are all equal to
2162:
2106:. The inverse or
2062:additive identity
2019:commutative rings
1787:is a set with an
1771:For example, the
1716:{\displaystyle g}
1533:is defined, then
1480:
1377:
1122:
1087:
1045:
930:doubling function
907:additive identity
788:is not defined.
681:is defined, then
605:
544:{\displaystyle *}
527:Identity elements
380:In this section,
343:binary operations
319:identity elements
297:
268:
231:is an invertible
221:inverse functions
5897:
5885:Abstract algebra
5840:
5793:
5792:
5785:
5779:
5776:
5770:
5767:
5761:
5742:
5693:
5691:
5690:
5685:
5683:
5682:
5617:
5616:
5571:
5570:
5517:
5516:
5513:
5494:
5492:
5491:
5486:
5484:
5479:
5471:
5468:
5459:
5458:
5402:
5394:
5389:
5388:
5347:
5346:
5299:
5291:
5282:
5281:
5273:
5269:
5268:
5267:
5264:
5249:
5248:
5245:
5235:
5234:
5193:
5185:
5180:
5179:
5171:
5170:
5127:
5126:
5118:
5114:
5113:
5112:
5109:
5090:
5089:
5051:
5050:
5003:
5002:
4950:
4949:
4946:
4923:
4921:
4920:
4915:
4910:
4909:
4901:
4897:
4896:
4895:
4892:
4877:
4876:
4873:
4863:
4855:
4852:
4828:
4826:
4825:
4820:
4818:
4817:
4735:
4733:
4732:
4727:
4719:
4718:
4715:
4709:
4708:
4700:
4696:
4692:
4691:
4688:
4664:and is given by
4655:
4653:
4652:
4647:
4623:
4621:
4620:
4615:
4613:
4612:
4599:
4597:
4589:
4586:
4579:
4574:
4573:
4572:
4564:
4560:
4559:
4558:
4555:
4540:
4539:
4536:
4516:
4514:
4513:
4508:
4470:
4468:
4467:
4462:
4460:
4459:
4443:
4441:
4433:
4430:
4423:
4418:
4417:
4416:
4413:
4407:
4406:
4398:
4394:
4390:
4389:
4386:
4363:
4361:
4360:
4355:
4308:
4306:
4305:
4300:
4284:
4282:
4281:
4276:
4264:commutative ring
4254:
4252:
4251:
4246:
4226:
4224:
4223:
4218:
4203:
4201:
4200:
4195:
3658:
3656:
3655:
3650:
3629:
3627:
3626:
3621:
3593:
3591:
3590:
3585:
3447:
3446:
3436:
3434:
3433:
3428:
3345:
3344:
3338:
3336:
3335:
3330:
3240:
3239:
3236:
3234:
3233:
3228:
3200:right invertible
3193:
3191:
3190:
3185:
3169:
3167:
3166:
3161:
3149:
3147:
3146:
3141:
3121:
3119:
3118:
3113:
3101:
3099:
3098:
3093:
3081:
3079:
3078:
3073:
3062:. If an element
3061:
3059:
3058:
3053:
3037:
3035:
3034:
3029:
3017:
3015:
3014:
3009:
2993:
2991:
2990:
2985:
2973:
2971:
2970:
2965:
2941:
2939:
2938:
2933:
2909:
2907:
2906:
2901:
2883:identity element
2880:
2878:
2877:
2872:
2860:binary operation
2849:
2847:
2846:
2841:
2784:inverse function
2762:
2758:
2756:
2755:
2750:
2736:
2735:
2713:
2705:
2703:
2702:
2697:
2681:
2673:
2671:
2670:
2665:
2649:
2647:
2646:
2641:
2636:
2615:
2613:
2612:
2607:
2599:
2584:
2582:
2581:
2576:
2559:the composition
2558:
2556:
2555:
2550:
2539:
2518:
2516:
2515:
2510:
2437:
2433:
2420:integer matrices
2413:
2398:
2394:
2386:
2375:
2371:
2355:
2345:
2341:
2331:
2327:
2323:
2319:
2309:
2291:commutative ring
2242:rational numbers
2235:
2226:commutative ring
2215:linear functions
2197:
2190:
2182:
2176:
2174:
2173:
2168:
2163:
2155:
2146:
2144:
2143:
2138:
2133:
2132:
2113:
2097:
2082:
2074:additive inverse
2071:
2067:
1999:rational numbers
1984:
1982:
1980:
1979:
1974:
1955:
1953:
1952:
1947:
1923:
1921:
1920:
1915:
1894:
1892:
1891:
1886:
1862:
1793:identity element
1768:
1766:
1765:
1760:
1745:
1743:
1742:
1737:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1692:
1691:
1619:, an invertible
1604:
1602:
1601:
1596:
1591:
1590:
1578:
1577:
1558:
1556:
1555:
1550:
1532:
1530:
1529:
1524:
1506:
1502:
1491:
1489:
1488:
1483:
1481:
1473:
1460:
1458:
1457:
1452:
1450:
1449:
1427:
1425:
1424:
1419:
1414:
1413:
1391:
1389:
1388:
1383:
1378:
1370:
1357:
1355:
1354:
1349:
1344:
1343:
1324:
1320:
1318:
1317:
1312:
1294:
1291:, of an element
1289:additive inverse
1277:
1275:
1274:
1269:
1197:
1193:
1189:
1141:is even or odd.
1140:
1136:
1134:
1133:
1128:
1123:
1118:
1107:
1098:
1096:
1095:
1090:
1088:
1080:
1071:
1063:
1061:
1060:
1055:
1050:
1046:
1038:
1019:
1017:
1016:
1011:
990:
988:
987:
982:
967:
959:
957:
956:
951:
915:additive inverse
912:
904:
893:
885:
881:
873:
869:
865:
863:
862:
857:
811:
804:
787:
785:
784:
779:
761:
757:
750:
748:
747:
742:
709:
707:
706:
701:
680:
678:
677:
672:
654:
650:
643:
639:
632:
630:
629:
624:
606:
603:
575:
565:identity element
558:
550:
548:
547:
542:
511:
507:
490:
488:
487:
482:
414:
412:
411:
406:
383:
375:identity element
333:The concepts of
309:
307:
306:
301:
299:
290:
280:
278:
277:
272:
270:
261:
217:inverse matrices
196:, also called a
174:additive inverse
163:
160:, if an element
155:
148:
144:
140:
126:
112:
104:
100:
92:
89:, one says that
88:
74:
69:identity element
66:
55:
44:
5905:
5904:
5900:
5899:
5898:
5896:
5895:
5894:
5870:
5869:
5864:Semigroup Forum
5857:Semigroup Forum
5851:Miyuki Yamada,
5837:
5827:Clarendon Press
5802:
5797:
5796:
5787:
5786:
5782:
5777:
5773:
5768:
5764:
5743:
5739:
5734:
5707:
5699:singular matrix
5677:
5676:
5671:
5666:
5660:
5659:
5654:
5649:
5643:
5642:
5637:
5632:
5622:
5621:
5611:
5610:
5605:
5600:
5594:
5593:
5588:
5583:
5573:
5572:
5565:
5564:
5559:
5553:
5552:
5547:
5541:
5540:
5535:
5525:
5524:
5512:
5508:
5506:
5503:
5502:
5482:
5481:
5472:
5467:
5453:
5452:
5444:
5438:
5437:
5432:
5423:
5422:
5417:
5404:
5403:
5393:
5383:
5382:
5377:
5368:
5367:
5359:
5349:
5348:
5341:
5340:
5335:
5329:
5328:
5323:
5317:
5316:
5311:
5301:
5300:
5290:
5283:
5274:
5263:
5259:
5255:
5251:
5250:
5244:
5240:
5237:
5236:
5229:
5228:
5223:
5214:
5213:
5205:
5195:
5194:
5184:
5172:
5165:
5164:
5159:
5153:
5152:
5147:
5137:
5136:
5135:
5128:
5119:
5108:
5104:
5100:
5096:
5095:
5092:
5091:
5084:
5083:
5078:
5072:
5071:
5066:
5056:
5055:
5045:
5044:
5039:
5033:
5032:
5027:
5021:
5020:
5015:
5005:
5004:
4997:
4996:
4991:
4986:
4980:
4979:
4974:
4969:
4959:
4958:
4951:
4945:
4941:
4934:
4932:
4929:
4928:
4902:
4891:
4887:
4883:
4879:
4878:
4872:
4868:
4856:
4851:
4845:
4842:
4841:
4812:
4811:
4806:
4801:
4795:
4794:
4789:
4784:
4774:
4773:
4753:
4750:
4749:
4714:
4710:
4701:
4687:
4683:
4682:
4678:
4677:
4669:
4666:
4665:
4632:
4629:
4628:
4608:
4604:
4590:
4585:
4580:
4565:
4554:
4550:
4546:
4542:
4541:
4535:
4531:
4530:
4528:
4522:
4519:
4518:
4478:
4475:
4474:
4455:
4451:
4434:
4429:
4424:
4412:
4408:
4399:
4385:
4381:
4380:
4376:
4375:
4374:
4372:
4369:
4366:
4365:
4325:
4322:
4321:
4294:
4291:
4290:
4270:
4267:
4266:
4240:
4237:
4236:
4212:
4209:
4208:
4189:
4186:
4185:
4179:
4143:
4135:
4130:
4124:
3932:
3902:
3890:
3878:
3871:
3688:In a semigroup
3678:
3672:
3665:
3635:
3632:
3631:
3615:
3612:
3611:
3579:
3576:
3575:
3410:
3407:
3406:
3312:
3309:
3308:
3210:
3207:
3206:
3196:left invertible
3179:
3176:
3175:
3155:
3152:
3151:
3135:
3132:
3131:
3126:, or simply an
3107:
3104:
3103:
3087:
3084:
3083:
3067:
3064:
3063:
3047:
3044:
3043:
3023:
3020:
3019:
3003:
3000:
2999:
2979:
2976:
2975:
2947:
2944:
2943:
2915:
2912:
2911:
2889:
2886:
2885:
2866:
2863:
2862:
2835:
2832:
2831:
2828:
2823:
2821:Generalizations
2760:
2731:
2727:
2725:
2722:
2721:
2711:
2691:
2688:
2687:
2679:
2659:
2656:
2655:
2629:
2621:
2618:
2617:
2592:
2590:
2587:
2586:
2564:
2561:
2560:
2532:
2524:
2521:
2520:
2492:
2489:
2488:
2448:
2435:
2431:
2418:In the case of
2411:
2396:
2392:
2384:
2373:
2369:
2358:identity matrix
2347:
2343:
2333:
2329:
2325:
2321:
2311:
2307:
2262:
2250:integral domain
2233:
2195:
2188:
2180:
2154:
2152:
2149:
2148:
2125:
2121:
2119:
2116:
2115:
2111:
2095:
2077:
2069:
2065:
2027:
2011:integral domain
1995:natural numbers
1962:
1959:
1958:
1957:
1929:
1926:
1925:
1900:
1897:
1896:
1868:
1865:
1864:
1860:
1781:
1751:
1748:
1747:
1728:
1725:
1724:
1708:
1705:
1704:
1684:
1680:
1678:
1675:
1674:
1671:transformations
1633:
1617:category theory
1583:
1579:
1570:
1566:
1564:
1561:
1560:
1538:
1535:
1534:
1512:
1509:
1508:
1504:
1500:
1472:
1470:
1467:
1466:
1439:
1435:
1433:
1430:
1429:
1403:
1399:
1397:
1394:
1393:
1369:
1367:
1364:
1363:
1362:multiplication
1336:
1332:
1330:
1327:
1326:
1322:
1300:
1297:
1296:
1292:
1206:
1203:
1202:
1195:
1191:
1187:
1177:
1161:category theory
1138:
1108:
1106:
1104:
1101:
1100:
1079:
1077:
1074:
1073:
1069:
1037:
1033:
1025:
1022:
1021:
996:
993:
992:
973:
970:
969:
965:
936:
933:
932:
919:natural numbers
910:
902:
891:
883:
879:
871:
867:
836:
833:
832:
829:
809:
806:identity matrix
796:
795:, there is one
767:
764:
763:
759:
755:
715:
712:
711:
686:
683:
682:
660:
657:
656:
652:
648:
641:
637:
602:
584:
581:
580:
573:
568:, or simply an
556:
536:
533:
532:
529:
509:
495:
434:
431:
430:
420:
397:
394:
393:
381:
335:inverse element
331:
326:
288:
286:
283:
282:
259:
257:
254:
253:
225:category theory
166:inverse element
161:
153:
146:
142:
128:
114:
110:
102:
98:
90:
76:
72:
64:
56:) of numbers.
50:
39:
32:inverse element
24:
17:
12:
11:
5:
5903:
5893:
5892:
5887:
5882:
5868:
5867:
5860:
5849:
5844:Drazin, M.P.,
5842:
5835:
5818:
5801:
5798:
5795:
5794:
5780:
5771:
5762:
5736:
5735:
5733:
5730:
5729:
5728:
5723:
5721:Loop (algebra)
5718:
5713:
5706:
5703:
5695:
5694:
5681:
5675:
5672:
5670:
5667:
5665:
5662:
5661:
5658:
5655:
5653:
5650:
5648:
5645:
5644:
5641:
5638:
5636:
5633:
5631:
5628:
5627:
5625:
5620:
5615:
5609:
5606:
5604:
5601:
5599:
5596:
5595:
5592:
5589:
5587:
5584:
5582:
5579:
5578:
5576:
5569:
5563:
5560:
5558:
5555:
5554:
5551:
5548:
5546:
5543:
5542:
5539:
5536:
5534:
5531:
5530:
5528:
5523:
5520:
5511:
5496:
5495:
5478:
5475:
5466:
5462:
5457:
5451:
5448:
5445:
5443:
5440:
5439:
5436:
5433:
5431:
5428:
5425:
5424:
5421:
5418:
5416:
5413:
5410:
5409:
5407:
5400:
5397:
5392:
5387:
5381:
5378:
5376:
5373:
5370:
5369:
5366:
5363:
5360:
5358:
5355:
5354:
5352:
5345:
5339:
5336:
5334:
5331:
5330:
5327:
5324:
5322:
5319:
5318:
5315:
5312:
5310:
5307:
5306:
5304:
5297:
5294:
5289:
5286:
5284:
5280:
5277:
5272:
5262:
5258:
5254:
5243:
5239:
5238:
5233:
5227:
5224:
5222:
5219:
5216:
5215:
5212:
5209:
5206:
5204:
5201:
5200:
5198:
5191:
5188:
5183:
5178:
5175:
5169:
5163:
5160:
5158:
5155:
5154:
5151:
5148:
5146:
5143:
5142:
5140:
5134:
5131:
5129:
5125:
5122:
5117:
5107:
5103:
5099:
5094:
5093:
5088:
5082:
5079:
5077:
5074:
5073:
5070:
5067:
5065:
5062:
5061:
5059:
5054:
5049:
5043:
5040:
5038:
5035:
5034:
5031:
5028:
5026:
5023:
5022:
5019:
5016:
5014:
5011:
5010:
5008:
5001:
4995:
4992:
4990:
4987:
4985:
4982:
4981:
4978:
4975:
4973:
4970:
4968:
4965:
4964:
4962:
4957:
4954:
4952:
4944:
4940:
4937:
4936:
4913:
4908:
4905:
4900:
4890:
4886:
4882:
4871:
4867:
4862:
4859:
4850:
4830:
4829:
4816:
4810:
4807:
4805:
4802:
4800:
4797:
4796:
4793:
4790:
4788:
4785:
4783:
4780:
4779:
4777:
4772:
4769:
4766:
4763:
4760:
4757:
4740:rank deficient
4725:
4722:
4713:
4707:
4704:
4699:
4695:
4686:
4681:
4676:
4673:
4645:
4642:
4639:
4636:
4625:
4624:
4611:
4607:
4603:
4596:
4593:
4584:
4577:
4571:
4568:
4563:
4553:
4549:
4545:
4534:
4526:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4482:
4471:
4458:
4454:
4450:
4447:
4440:
4437:
4428:
4421:
4411:
4405:
4402:
4397:
4393:
4384:
4379:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4298:
4287:if and only if
4285:is invertible
4274:
4244:
4216:
4193:
4178:
4175:
4142:
4139:
4134:
4131:
4126:Main article:
4123:
4120:
4024:
4023:
3995:
3950:; this endows
3931:
3925:
3900:
3888:
3876:
3869:
3674:Main article:
3671:
3670:In a semigroup
3668:
3663:
3648:
3645:
3642:
3639:
3619:
3608:group of units
3583:
3567:
3566:
3563:
3560:
3557:
3554:
3551:
3547:
3546:
3543:
3540:
3537:
3534:
3531:
3527:
3526:
3523:
3520:
3517:
3514:
3511:
3507:
3506:
3503:
3500:
3497:
3494:
3491:
3487:
3486:
3483:
3480:
3477:
3474:
3471:
3467:
3466:
3463:
3460:
3457:
3454:
3451:
3426:
3423:
3420:
3417:
3414:
3401:
3400:
3397:
3394:
3391:
3387:
3386:
3383:
3380:
3377:
3373:
3372:
3369:
3366:
3363:
3359:
3358:
3355:
3352:
3349:
3328:
3325:
3322:
3319:
3316:
3296:
3295:
3292:
3289:
3286:
3282:
3281:
3278:
3275:
3272:
3268:
3267:
3264:
3261:
3258:
3254:
3253:
3250:
3247:
3244:
3226:
3223:
3220:
3217:
3214:
3183:
3159:
3139:
3111:
3091:
3071:
3051:
3027:
3007:
2983:
2963:
2960:
2957:
2954:
2951:
2931:
2928:
2925:
2922:
2919:
2899:
2896:
2893:
2870:
2839:
2827:
2824:
2822:
2819:
2748:
2745:
2742:
2739:
2734:
2730:
2695:
2663:
2639:
2635:
2632:
2628:
2625:
2605:
2602:
2598:
2595:
2574:
2571:
2568:
2548:
2545:
2542:
2538:
2535:
2531:
2528:
2508:
2505:
2502:
2499:
2496:
2447:
2444:
2401:inverse matrix
2346:is defined if
2328:columns), and
2293:are considered
2261:
2258:
2166:
2161:
2158:
2136:
2131:
2128:
2124:
2064:, and denoted
2043:multiplication
2026:
2023:
1972:
1969:
1966:
1945:
1942:
1939:
1936:
1933:
1913:
1910:
1907:
1904:
1884:
1881:
1878:
1875:
1872:
1819:group of units
1780:
1777:
1758:
1755:
1735:
1732:
1712:
1690:
1687:
1683:
1632:
1629:
1607:An invertible
1594:
1589:
1586:
1582:
1576:
1573:
1569:
1548:
1545:
1542:
1522:
1519:
1516:
1479:
1476:
1448:
1445:
1442:
1438:
1417:
1412:
1409:
1406:
1402:
1381:
1376:
1373:
1347:
1342:
1339:
1335:
1310:
1307:
1304:
1279:
1278:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1179:An element is
1176:
1173:
1126:
1121:
1117:
1114:
1111:
1086:
1083:
1066:floor function
1053:
1049:
1044:
1041:
1036:
1032:
1029:
1009:
1006:
1003:
1000:
980:
977:
949:
946:
943:
940:
855:
852:
849:
846:
843:
840:
828:
825:
777:
774:
771:
740:
737:
734:
731:
728:
725:
722:
719:
699:
696:
693:
690:
670:
667:
664:
634:
633:
622:
619:
616:
613:
610:
600:
597:
594:
591:
588:
572:is an element
551:be a possibly
540:
528:
525:
492:
491:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
419:
416:
404:
401:
330:
327:
296:
293:
267:
264:
168:or simply the
15:
9:
6:
4:
3:
2:
5902:
5891:
5888:
5886:
5883:
5881:
5878:
5877:
5875:
5865:
5861:
5858:
5854:
5850:
5847:
5843:
5838:
5836:0-19-851194-9
5832:
5828:
5824:
5819:
5816:
5815:3-11-015248-7
5812:
5808:
5804:
5803:
5790:
5784:
5775:
5766:
5759:
5755:
5751:
5747:
5741:
5737:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5711:Division ring
5709:
5708:
5702:
5700:
5679:
5673:
5668:
5663:
5656:
5651:
5646:
5639:
5634:
5629:
5623:
5618:
5613:
5607:
5602:
5597:
5590:
5585:
5580:
5574:
5567:
5561:
5556:
5549:
5544:
5537:
5532:
5526:
5521:
5518:
5509:
5501:
5500:
5499:
5476:
5473:
5464:
5460:
5455:
5449:
5446:
5441:
5434:
5429:
5426:
5419:
5414:
5411:
5405:
5398:
5395:
5390:
5385:
5379:
5374:
5371:
5364:
5361:
5356:
5350:
5343:
5337:
5332:
5325:
5320:
5313:
5308:
5302:
5295:
5292:
5287:
5285:
5278:
5275:
5270:
5260:
5256:
5252:
5241:
5231:
5225:
5220:
5217:
5210:
5207:
5202:
5196:
5189:
5186:
5181:
5176:
5173:
5167:
5161:
5156:
5149:
5144:
5138:
5132:
5130:
5123:
5120:
5115:
5105:
5101:
5097:
5086:
5080:
5075:
5068:
5063:
5057:
5052:
5047:
5041:
5036:
5029:
5024:
5017:
5012:
5006:
4999:
4993:
4988:
4983:
4976:
4971:
4966:
4960:
4955:
4953:
4942:
4938:
4927:
4926:
4925:
4911:
4906:
4903:
4898:
4888:
4884:
4880:
4869:
4865:
4860:
4857:
4848:
4839:
4835:
4814:
4808:
4803:
4798:
4791:
4786:
4781:
4775:
4770:
4767:
4764:
4761:
4758:
4755:
4748:
4747:
4746:
4743:
4741:
4736:
4723:
4720:
4711:
4705:
4702:
4697:
4693:
4684:
4679:
4674:
4671:
4663:
4659:
4658:least squares
4643:
4640:
4637:
4634:
4609:
4605:
4601:
4594:
4591:
4582:
4575:
4569:
4566:
4561:
4551:
4547:
4543:
4532:
4524:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4472:
4456:
4452:
4448:
4445:
4438:
4435:
4426:
4419:
4409:
4403:
4400:
4395:
4391:
4382:
4377:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4319:
4318:
4317:
4315:
4310:
4296:
4288:
4272:
4265:
4260:
4258:
4242:
4234:
4230:
4214:
4207:
4191:
4184:
4183:square matrix
4174:
4172:
4168:
4164:
4160:
4156:
4152:
4148:
4138:
4129:
4119:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4021:
4017:
4013:
4009:
4005:
4001:
4000:
3996:
3994:
3990:
3986:
3982:
3980:
3976:
3975:
3974:
3972:
3968:
3964:
3960:
3958:
3953:
3949:
3945:
3941:
3937:
3929:
3924:
3922:
3918:
3917:quasi-inverse
3913:
3911:
3907:
3899:
3895:
3891:
3887:
3881:
3879:
3872:
3865:
3861:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3829:
3825:
3821:
3817:
3813:
3809:
3805:
3801:
3797:
3793:
3789:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3729:
3726:. An element
3725:
3724:
3723:pseudoinverse
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3686:
3684:
3677:
3667:
3662:
3643:
3637:
3617:
3609:
3606:, called the
3605:
3601:
3597:
3581:
3572:
3564:
3561:
3558:
3555:
3552:
3549:
3548:
3544:
3541:
3538:
3535:
3532:
3529:
3528:
3524:
3521:
3518:
3515:
3512:
3509:
3508:
3504:
3501:
3498:
3495:
3492:
3489:
3488:
3484:
3481:
3478:
3475:
3472:
3469:
3468:
3448:
3445:
3442:
3440:
3421:
3418:
3415:
3398:
3395:
3392:
3389:
3388:
3384:
3381:
3378:
3375:
3374:
3370:
3367:
3364:
3361:
3360:
3346:
3343:
3342:
3339:given by the
3323:
3320:
3317:
3306:
3301:
3293:
3290:
3287:
3284:
3283:
3279:
3276:
3273:
3270:
3269:
3265:
3262:
3259:
3256:
3255:
3241:
3238:
3221:
3218:
3215:
3203:
3201:
3197:
3181:
3173:
3157:
3137:
3129:
3125:
3109:
3089:
3069:
3049:
3041:
3040:right inverse
3025:
3005:
2997:
2981:
2961:
2958:
2955:
2952:
2949:
2929:
2926:
2923:
2920:
2917:
2897:
2894:
2891:
2884:
2868:
2861:
2857:
2854:, that is, a
2853:
2837:
2818:
2816:
2812:
2808:
2804:
2803:vector spaces
2800:
2796:
2791:
2789:
2785:
2781:
2776:
2774:
2770:
2766:
2746:
2740:
2737:
2732:
2728:
2720:
2715:
2709:
2693:
2685:
2677:
2661:
2653:
2637:
2633:
2630:
2626:
2623:
2603:
2600:
2596:
2593:
2572:
2569:
2566:
2546:
2543:
2536:
2533:
2529:
2526:
2506:
2500:
2497:
2494:
2485:
2483:
2478:
2476:
2472:
2468:
2464:
2460:
2459:homomorphisms
2456:
2452:
2443:
2439:
2429:
2425:
2421:
2416:
2408:
2406:
2405:Cramer's rule
2402:
2390:
2382:
2377:
2367:
2366:main diagonal
2363:
2362:square matrix
2359:
2354:
2350:
2340:
2336:
2318:
2314:
2304:
2302:
2298:
2294:
2292:
2286:
2282:
2278:
2274:
2270:
2266:
2257:
2255:
2251:
2247:
2243:
2239:
2231:
2230:zero divisors
2227:
2222:
2220:
2216:
2212:
2207:
2205:
2204:division ring
2201:
2192:
2186:
2177:
2164:
2159:
2156:
2134:
2129:
2126:
2122:
2109:
2105:
2101:
2093:
2089:
2084:
2081:
2076:and denoted
2075:
2063:
2059:
2055:
2051:
2050:abelian group
2046:
2044:
2040:
2036:
2032:
2022:
2020:
2016:
2015:localizations
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1970:
1967:
1964:
1943:
1940:
1937:
1934:
1931:
1911:
1908:
1905:
1902:
1882:
1879:
1876:
1873:
1870:
1858:
1852:
1850:
1846:
1842:
1838:
1834:
1829:
1827:
1822:
1821:of the ring.
1820:
1817:and form the
1816:
1812:
1807:
1805:
1801:
1796:
1794:
1790:
1786:
1776:
1774:
1769:
1756:
1753:
1733:
1730:
1710:
1688:
1685:
1681:
1672:
1668:
1663:
1661:
1657:
1653:
1648:
1646:
1642:
1638:
1628:
1626:
1622:
1618:
1614:
1611:is called an
1610:
1605:
1592:
1587:
1584:
1580:
1574:
1571:
1567:
1546:
1543:
1540:
1520:
1517:
1514:
1497:
1495:
1477:
1474:
1464:
1446:
1443:
1440:
1436:
1428:The notation
1415:
1410:
1407:
1404:
1400:
1379:
1374:
1371:
1361:
1345:
1340:
1337:
1333:
1308:
1305:
1302:
1290:
1285:
1283:
1265:
1262:
1259:
1256:
1253:
1247:
1244:
1241:
1235:
1229:
1226:
1223:
1217:
1214:
1211:
1208:
1201:
1200:
1199:
1184:
1182:
1172:
1170:
1166:
1162:
1157:
1155:
1151:
1147:
1142:
1124:
1119:
1115:
1112:
1109:
1084:
1081:
1067:
1051:
1047:
1042:
1039:
1034:
1027:
1007:
1004:
1001:
998:
978:
975:
963:
947:
944:
938:
931:
927:
922:
920:
916:
908:
900:
895:
889:
888:right inverse
877:
853:
850:
847:
844:
841:
838:
824:
822:
818:
813:
807:
803:
799:
794:
789:
775:
772:
769:
752:
738:
735:
732:
729:
726:
723:
720:
717:
697:
694:
691:
688:
668:
665:
662:
645:
620:
617:
614:
611:
608:
598:
595:
592:
589:
586:
579:
578:
577:
571:
567:
566:
560:
554:
538:
524:
522:
518:
513:
506:
502:
498:
478:
475:
469:
466:
463:
457:
451:
448:
445:
439:
436:
429:
428:
427:
425:
418:Associativity
415:
402:
399:
391:
387:
378:
376:
372:
371:associativity
368:
364:
360:
356:
352:
348:
344:
340:
336:
324:
320:
316:
311:
294:
291:
265:
262:
251:
247:
246:
241:
236:
234:
230:
226:
222:
218:
214:
210:
206:
201:
199:
195:
191:
187:
183:
179:
175:
171:
167:
159:
150:
139:
135:
131:
125:
121:
117:
108:
107:right inverse
96:
87:
83:
79:
70:
63:denoted here
62:
57:
54:
48:
43:
37:
33:
29:
22:
5852:
5845:
5822:
5806:
5783:
5778:Howie p. 102
5774:
5765:
5740:
5696:
5497:
4837:
4833:
4831:
4744:
4737:
4660:formula for
4626:
4311:
4261:
4180:
4170:
4166:
4162:
4158:
4154:
4150:
4144:
4136:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4051:
4047:
4043:
4039:
4035:
4034:; these are
4027:
4025:
4019:
4011:
4007:
4003:
3999:*-semigroups
3997:
3992:
3988:
3984:
3978:
3977:
3970:
3966:
3962:
3956:
3955:
3951:
3947:
3943:
3939:
3935:
3933:
3927:
3916:
3914:
3897:
3893:
3885:
3882:
3874:
3867:
3863:
3855:
3851:
3847:
3843:
3839:
3835:
3831:
3827:
3823:
3819:
3815:
3811:
3807:
3803:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3735:
3731:
3727:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3687:
3679:
3660:
3573:
3570:
3443:
3439:Latin square
3404:
3341:Cayley table
3302:
3299:
3204:
3199:
3195:
3171:
3127:
3123:
3122:is called a
3039:
3038:is called a
2996:left inverse
2995:
2994:is called a
2850:be a unital
2829:
2805:but not for
2792:
2787:
2777:
2718:
2717:There is an
2716:
2486:
2479:
2474:
2449:
2440:
2428:real numbers
2417:
2409:
2378:
2352:
2348:
2338:
2334:
2316:
2312:
2305:
2288:
2263:
2238:localization
2223:
2208:
2193:
2187:, which has
2178:
2099:
2094:and denoted
2085:
2079:
2047:
2042:
2038:
2028:
1859:(an element
1853:
1830:
1823:
1808:
1799:
1797:
1791:that has an
1782:
1770:
1669:on a set as
1665:A group may
1664:
1649:
1634:
1609:homomorphism
1606:
1498:
1286:
1281:
1280:
1185:
1180:
1178:
1158:
1143:
929:
923:
901:, which has
896:
887:
876:left inverse
875:
830:
814:
801:
797:
790:
753:
646:
635:
569:
563:
561:
530:
514:
504:
500:
496:
493:
421:
390:proper class
388:(possibly a
379:
338:
334:
332:
237:
202:
193:
185:
169:
165:
151:
137:
133:
129:
123:
119:
115:
106:
95:left inverse
94:
85:
81:
77:
58:
52:
41:
31:
25:
5697:which is a
4233:determinant
3981:-semigroups
3930:-semigroups
3800:idempotents
3692:an element
3596:associative
2706:equals the
2482:associative
2475:composition
2451:Composition
2301:determinant
2287:. However,
2221:to itself.
2206:otherwise.
2114:is denoted
2058:associative
2054:commutative
1826:commutative
1625:isomorphism
1613:isomorphism
1360:commutative
1295:is denoted
1282:The inverse
1169:retractions
815:Similarly,
576:such that
424:associative
315:associative
229:isomorphism
158:associative
101:, and that
28:mathematics
5874:Categories
5800:References
4662:regression
4259:for more.
4078:such that
4016:involution
3959:-semigroup
3802:, that is
3708:such that
3696:is called
3172:invertible
3170:is called
2942:, we have
2910:. If, for
2799:surjective
2471:categories
1779:In monoids
1660:involution
1181:invertible
1154:surjective
1068:that maps
968:to either
636:for every
494:for every
47:reciprocal
5474:−
5447:−
5427:−
5412:−
5372:−
5362:−
5276:−
5218:−
5208:−
5174:−
5121:−
4904:−
4858:−
4765:×
4703:−
4592:−
4576:⏟
4567:−
4496:∣
4490:×
4436:−
4420:⏟
4401:−
4343:∣
4337:×
4314:full rank
4122:Semirings
3683:semigroup
3582:∗
3422:∗
3324:∗
3222:∗
2953:∗
2927:∈
2895:∈
2869:∗
2795:injective
2780:bijection
2744:→
2738::
2627:⊂
2570:∘
2541:→
2530::
2504:→
2498::
2467:morphisms
2324:rows and
2285:semirings
2185:zero ring
2127:−
1833:functions
1686:−
1631:In groups
1585:−
1572:−
1544:∗
1518:∗
1444:−
1441:∘
1408:−
1405:∗
1338:−
1303:−
1254:∗
1245:∗
1227:∗
1218:∗
1150:injective
1113:−
1031:↦
942:↦
926:functions
842:∗
773:∗
727:∗
666:∗
612:∗
590:∗
539:∗
476:∗
467:∗
449:∗
440:∗
400:∗
363:morphisms
317:and have
250:fractions
67:, and an
61:operation
59:Given an
5705:See also
4133:Examples
3942:for all
3850:, while
2719:identity
2684:codomain
2652:codomain
2634:′
2597:′
2537:′
2269:matrices
2260:Matrices
2039:addition
2025:In rings
2003:integers
1991:integers
1956:implies
1895:implies
1652:function
1643:with an
1621:morphism
1465:, since
1198:, then
1175:Inverses
1165:sections
1048:⌋
1035:⌊
570:identity
367:category
245:inversus
233:morphism
141:for all
71:denoted
36:opposite
5880:Algebra
4832:So, as
3732:inverse
3128:inverse
3102:, then
2974:, then
2881:and an
2858:with a
2807:modules
2271:over a
2254:subring
2217:from a
1615:. In
553:partial
170:inverse
5833:
5813:
5752:, and
3938:°)° =
3842:, and
3600:monoid
2769:object
2708:domain
2676:domain
2248:of an
2088:monoid
2033:is an
2013:, and
2009:of an
1785:monoid
909:, and
882:, and
866:where
347:domain
205:groups
180:, and
45:) and
5732:Notes
5469:right
4853:right
4836:<
4587:right
4206:field
4106:)* =
4098:)* =
4094:, (
4006:)° =
3782:then
3604:group
3130:, of
2852:magma
2813:or a
2453:is a
2332:is a
2310:is a
2277:rings
2273:field
2209:In a
2200:field
2098:. An
2001:from
1993:from
1815:units
1804:group
1656:arity
1639:is a
1637:group
886:is a
874:is a
384:is a
365:in a
240:Latin
213:units
209:rings
192:, an
105:is a
93:is a
75:, if
5831:ISBN
5811:ISBN
4502:<
4473:For
4431:left
4349:>
4320:For
4165:and
4153:and
4042:° =
3810:and
3798:are
3790:and
3746:and
3305:loop
3018:and
2830:Let
2519:and
2465:and
2391:in
2389:unit
2299:and
2297:rank
2283:and
2281:rngs
2104:unit
2056:and
2041:and
2031:ring
1924:and
1811:ring
1798:The
1503:and
1190:and
1064:the
758:and
651:and
640:and
531:Let
426:if
373:and
337:and
219:and
198:unit
190:ring
145:and
127:and
4738:No
4167:GLG
4159:LGL
4114:or
4102:, (
4088:yxy
4080:xyx
3946:in
3923:).
3768:zxz
3760:xzx
3752:yxy
3740:xyx
3738:if
3734:of
3710:xzx
3704:in
3659:or
3610:of
3594:is
3198:or
3174:in
3042:of
2998:of
2856:set
2797:or
2765:set
2710:of
2686:of
2678:of
2654:of
2487:If
2469:of
2461:of
2434:or
2410:If
2407:.
2379:An
2306:If
2194:If
2017:of
1667:act
1641:set
1499:If
1159:In
1156:.
1099:or
1072:to
991:or
905:as
890:of
878:of
831:If
647:If
604:and
562:An
508:in
386:set
310:).
281:is
156:is
109:of
97:of
26:In
5876::
5855:,
5829:.
5825:.
5674:45
5669:36
5664:27
5657:36
5652:29
5647:22
5640:27
5635:22
5630:17
5442:13
5415:17
5399:18
5380:14
5375:32
5365:32
5357:77
5296:54
5226:14
5221:32
5211:32
5203:77
5190:54
5162:77
5157:32
5150:32
5145:14
5081:77
5076:32
5069:32
5064:14
4309:.
4181:A
4169:=
4161:=
4149:,
4118:.
4108:yx
4104:yx
4100:xy
4096:xy
4090:=
4086:,
4082:=
4040:aa
4004:ab
3991:=
3985:aa
3838:=
3836:fy
3834:=
3832:ye
3830:,
3826:=
3824:xf
3822:=
3820:ex
3814:=
3812:ff
3806:=
3804:ee
3796:yx
3794:=
3788:xy
3786:=
3766:=
3758:=
3750:=
3742:=
3716:;
3712:=
3685:.
3666:.
3565:3
3562:2
3559:4
3556:1
3553:5
3550:5
3545:1
3542:3
3539:2
3536:5
3533:4
3530:4
3525:2
3522:1
3519:5
3516:4
3513:3
3510:3
3505:4
3502:5
3499:1
3496:3
3493:2
3490:2
3485:5
3482:4
3479:3
3476:2
3473:1
3470:1
3465:5
3462:4
3459:3
3456:2
3453:1
3450:*
3441:.
3399:1
3396:2
3393:3
3390:3
3385:2
3382:1
3379:2
3376:2
3371:3
3368:2
3365:1
3362:1
3357:3
3354:2
3351:1
3348:*
3294:1
3291:1
3288:3
3285:3
3280:1
3277:1
3274:2
3271:2
3266:3
3263:2
3260:1
3257:1
3252:3
3249:2
3246:1
3243:*
3202:.
2790:.
2729:id
2714:.
2484:.
2436:−1
2376:.
2351:=
2344:AB
2303:.
2279:,
2224:A
2029:A
2021:.
1997:,
1983:).
1851:.
1809:A
1795:.
1783:A
1635:A
1627:.
1496:.
1171:.
894:.
751:)
559:.
523:.
503:,
499:,
357:,
242::
235:.
176:,
136:=
132:*
122:=
118:*
84:=
80:∗
51:1/
5839:.
5791:.
5760:.
5680:]
5624:[
5619:=
5614:]
5608:6
5603:5
5598:4
5591:3
5586:2
5581:1
5575:[
5568:]
5562:6
5557:3
5550:5
5545:2
5538:4
5533:1
5527:[
5522:=
5519:A
5514:T
5510:A
5477:1
5465:A
5461:=
5456:]
5450:4
5435:2
5430:2
5420:8
5406:[
5396:1
5391:=
5386:]
5351:[
5344:]
5338:6
5333:3
5326:5
5321:2
5314:4
5309:1
5303:[
5293:1
5288:=
5279:1
5271:)
5265:T
5261:A
5257:A
5253:(
5246:T
5242:A
5232:]
5197:[
5187:1
5182:=
5177:1
5168:]
5139:[
5133:=
5124:1
5116:)
5110:T
5106:A
5102:A
5098:(
5087:]
5058:[
5053:=
5048:]
5042:6
5037:3
5030:5
5025:2
5018:4
5013:1
5007:[
5000:]
4994:6
4989:5
4984:4
4977:3
4972:2
4967:1
4961:[
4956:=
4947:T
4943:A
4939:A
4912:.
4907:1
4899:)
4893:T
4889:A
4885:A
4881:(
4874:T
4870:A
4866:=
4861:1
4849:A
4838:n
4834:m
4815:]
4809:6
4804:5
4799:4
4792:3
4787:2
4782:1
4776:[
4771:=
4768:3
4762:2
4759::
4756:A
4724:.
4721:b
4716:T
4712:A
4706:1
4698:)
4694:A
4689:T
4685:A
4680:(
4675:=
4672:x
4644:b
4641:=
4638:x
4635:A
4610:m
4606:I
4602:=
4595:1
4583:A
4570:1
4562:)
4556:T
4552:A
4548:A
4544:(
4537:T
4533:A
4525:A
4505:n
4499:m
4493:n
4487:m
4484::
4481:A
4457:n
4453:I
4449:=
4446:A
4439:1
4427:A
4414:T
4410:A
4404:1
4396:)
4392:A
4387:T
4383:A
4378:(
4352:n
4346:m
4340:n
4334:m
4331::
4328:A
4297:R
4273:R
4243:M
4215:K
4192:M
4171:G
4163:L
4155:G
4151:L
4092:y
4084:x
4076:y
4072:x
4068:a
4052:a
4048:a
4046:°
4044:a
4036:I
4028:I
4022:*
4020:a
4012:a
4010:°
4008:b
3993:a
3989:a
3987:°
3979:I
3971:U
3967:a
3963:a
3957:U
3952:S
3948:S
3944:a
3940:a
3936:a
3928:U
3901:e
3898:H
3894:e
3889:1
3886:H
3877:e
3875:L
3870:e
3868:R
3864:e
3856:y
3852:f
3848:x
3844:e
3840:y
3828:x
3816:f
3808:e
3792:f
3784:e
3780:x
3776:y
3772:x
3764:y
3756:x
3748:y
3744:x
3736:x
3728:y
3718:z
3714:x
3706:S
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3690:S
3664:1
3661:H
3647:)
3644:S
3641:(
3638:U
3618:S
3425:)
3419:,
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3413:(
3327:)
3321:,
3318:S
3315:(
3225:)
3219:,
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3213:(
3182:S
3158:S
3138:y
3110:x
3090:y
3070:x
3050:a
3026:b
3006:b
2982:a
2962:e
2959:=
2956:b
2950:a
2930:S
2924:b
2921:,
2918:a
2898:S
2892:e
2838:S
2763:(
2761:X
2747:X
2741:X
2733:X
2712:g
2694:f
2680:g
2662:f
2638:.
2631:Y
2624:Y
2604:Y
2601:=
2594:Y
2573:f
2567:g
2547:,
2544:Z
2534:Y
2527:g
2507:Y
2501:X
2495:f
2432:1
2412:R
2397:R
2393:R
2385:R
2374:0
2370:1
2353:p
2349:n
2339:q
2337:×
2335:p
2330:B
2326:n
2322:m
2317:n
2315:×
2313:m
2308:A
2234:0
2196:0
2189:0
2181:0
2165:.
2160:x
2157:1
2135:,
2130:1
2123:x
2112:x
2096:1
2080:x
2078:−
2070:x
2066:0
1971:z
1968:=
1965:y
1944:x
1941:z
1938:=
1935:x
1932:y
1912:,
1909:z
1906:=
1903:y
1883:z
1880:x
1877:=
1874:y
1871:x
1861:x
1757:.
1754:g
1734:,
1731:g
1711:g
1689:1
1682:g
1593:.
1588:1
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1575:1
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1521:y
1515:x
1505:y
1501:x
1478:f
1475:1
1447:1
1437:f
1416:.
1411:1
1401:x
1380:.
1375:x
1372:1
1346:,
1341:1
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1323:x
1309:.
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1293:x
1266:.
1263:r
1260:=
1257:r
1251:)
1248:x
1242:l
1239:(
1236:=
1233:)
1230:r
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1221:(
1215:l
1212:=
1209:l
1196:x
1192:r
1188:l
1139:n
1125:,
1120:2
1116:1
1110:n
1085:2
1082:n
1070:n
1052:,
1043:2
1040:n
1028:n
1008:1
1005:+
1002:n
999:2
979:n
976:2
966:n
948:x
945:2
939:x
911:0
903:0
892:x
884:y
880:y
872:x
868:e
854:,
851:e
848:=
845:y
839:x
810:n
802:n
800:×
798:n
776:f
770:e
760:f
756:e
739:.
736:f
733:=
730:f
724:e
721:=
718:e
698:.
695:f
692:=
689:e
669:f
663:e
653:f
649:e
642:y
638:x
621:y
618:=
615:y
609:e
599:x
596:=
593:e
587:x
574:e
557:X
510:X
505:z
501:y
497:x
479:z
473:)
470:y
464:x
461:(
458:=
455:)
452:z
446:y
443:(
437:x
403:.
382:X
325:.
295:x
292:y
266:y
263:x
162:x
154:∗
147:y
143:x
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134:y
130:e
124:x
120:e
116:x
111:x
103:y
99:y
91:x
86:e
82:y
78:x
73:e
65:∗
53:x
49:(
42:x
40:−
38:(
23:.
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