5504:
4941:
5499:{\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}}}
5703:
3914:
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
2452:
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
5515:
3691:
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
4633:
4480:
2812:, 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
4838:
2425:
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.
5698:{\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}}}
4933:
1865:
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
3930:. 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
4946:
1073:
4745:
2793:. 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
2768:
642:
4526:
4373:
2568:
1287:
2659:
4531:
4378:
500:
319:
290:
4753:
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.
2528:
1146:
1614:
2186:
1470:
1401:
2224:(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
1501:
1108:
969:
3894:
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
2625:
2951:
2594:
1437:
760:
2156:
1367:
3980:, 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
2919:
1712:
875:
3446:
3348:
3246:
2983:
1965:
1904:
4665:
3668:
1933:
1029:
719:
3891:. 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.
1992:
1568:
1542:
1330:
797:
690:
1778:
1755:
1000:
424:
4318:
4294:
4264:
4236:
4213:
3639:
3603:
3203:
3179:
3159:
3131:
3111:
3091:
3071:
3047:
3027:
3003:
2890:
2859:
2715:
2683:
1732:
560:
4762:
1786:
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.
2433:(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
908:
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
4854:
4266:
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
3613:(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
1839:, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).
935:
An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the
523:
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.
4121:. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the
2248:
2025:
975:, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps
4678:
1034:
5845:
5825:
2229:
1403:
When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in
1197:
In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if
211:, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).
5799:
2094:. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.
2734:
4042:
1681:
1179:
593:
4066:
31:
5900:
4074:
2239:(that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not
5895:
4070:
4009:
1175:
263:, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of
4026:
2779:
1670:
175:
has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the
4487:
4334:
2102:
71:
3945:
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (
2820:
over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a
2481:
2118:
1843:
1662:
1504:
1215:
936:
377:
260:
188:
57:
4628:{\displaystyle A\underbrace {A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}} _{A_{\text{right}}^{-1}}=I_{m}}
3248:
may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
5768:
4475:{\displaystyle \underbrace {\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}} _{A_{\text{left}}^{-1}}A=I_{n}}
4239:
2307:
2275:
1867:
1799:
1655:
803:
527:
365:
3455:
Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table
1783:
3919:. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an
2533:
5760:
4138:
3895:
3870:
3609:
then if an element has both a left inverse and a right inverse, they are equal. In other words, in a
2817:
2718:
2686:
2461:
2279:
1847:
1473:
1156:
972:
831:
531:
443:
369:
357:
295:
266:
226:. They are also commonly used for operations that are not defined for all possible operands, such as
2501:
4672:
4216:
3931:
3733:
3614:
2473:
2283:
2221:
2210:
2045:
1859:
1814:
1647:
1573:
1113:
909:
215:
5800:"MIT Professor Gilbert Strang Linear Algebra Lecture #33 – Left and Right Inverses; Pseudoinverse"
4184:
and one uniquely determines the other. They are not left or right inverses of each other however.
2630:
1442:
5736:
3810:
2825:
2399:
2287:
2256:
2161:
2114:
2041:
2017:
1997:
1855:
1851:
1836:
1825:
1821:
1376:
945:
223:
219:
208:
200:
192:
3965:
with a type ⟨2,1⟩ algebra. A semigroup endowed with such an operation is called a
1479:
1086:
5841:
5821:
5764:
5756:
4267:
4157:
3920:
3916:
3686:
2924:
2866:
2821:
2783:
2775:
2573:
2465:
2434:
2391:
2072:
1651:
1406:
917:
827:
724:
563:
396:
361:
353:
2263:. Localization is also used with zero divisors, but, in this case the original ring is not a
2128:
1339:
5890:
4274:
4073:(in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the
3926:
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a
2898:
2893:
2870:
2794:
2453:
case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.
2430:
2415:
2301:
2236:
2084:
2029:
1803:
1687:
1299:
925:
845:
575:
385:
329:
231:
227:
184:
79:
46:
3419:
3321:
3219:
2956:
1938:
1877:
1824:
is a monoid for ring multiplication. In this case, the invertible elements are also called
356:
that are everywhere defined (that is, the operation is defined for any two elements of its
5874:
5867:
5837:
5709:
4641:
3644:
2862:
2599:
2368:
2260:
2252:
2225:
2021:
2009:
1909:
1627:
1171:
1005:
816:
695:
235:
1971:
1547:
1521:
1309:
776:
669:
259:
that means 'turned upside down', 'overturned'. This may take its origin from the case of
1760:
1737:
982:
406:
5731:
4750:
4303:
4297:
4279:
4249:
4221:
4198:
3624:
3618:
3588:
3315:
3188:
3164:
3144:
3116:
3096:
3076:
3056:
3032:
3012:
2988:
2875:
2844:
2700:
2668:
2411:
2394:
is an invertible element under matrix multiplication. A matrix over a commutative ring
2005:
1829:
1717:
1076:
929:
834:, and the composition of the identity functions of two different sets are not defined.
545:
250:
4833:{\displaystyle A:2\times 3={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}}
4041:-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are
5884:
5721:
4668:
4193:
3829:. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and
2376:
2372:
2291:
2240:
2214:
2060:
388:
must be extended to partial operations; this is the object of the first subsections.
381:
17:
5755:
The usual definition of an identity element has been generalized for including the
3449:
3351:
2813:
2469:
1677:
1619:
400:
4928:{\displaystyle A_{\text{right}}^{-1}=A^{\text{T}}\left(AA^{\text{T}}\right)^{-1}.}
403:) on which a partial operation (possibly total) is defined, which is denoted with
4243:
3915:
exactly one inverse as defined in this section, then the semigroup is called an
3606:
2492:
2438:
2311:
2267:
of the localisation; instead, it is mapped non-injectively to the localization.
2068:
2064:
1635:
1623:
1370:
434:
325:
239:
168:
38:
2101:; this means that multiplication is associative and has an identity called the
5726:
3903:
have an inverse from the unital magma perspective, whereas for any idempotent
2809:
1734:
defines a transformation that is the inverse of the transformation defined by
1164:
823:, and two identity matrices of different size cannot be multiplied together.
5818:
Monoids, Acts and
Categories with Applications to Wreath Products and Graphs
4324:
3693:
2805:
2790:
2195:
1160:
255:
1658:
that has an identity element, and for which every element has an inverse.
765:
It follows that a total operation has at most one identity element, and if
5852:
contains all of the semigroup material herein except *-regular semigroups.
5820:, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000,
2441:. A square integer matrix is unimodular if and only if its determinant is
364:, that is operations that are not defined everywhere. Common examples are
4238:
is invertible (in the set of all square matrices of the same size, under
2694:
2662:
2477:
2295:
1631:
373:
243:
4065:°. There are few concrete examples of such semigroups however; most are
2000:
construction. This is the method that is commonly used for constructing
1757:
that is, the transformation that "undoes" the transformation defined by
1673:, since the inverse of the inverse of an element is the element itself.
1665:
from the group to itself that may also be considered as an operation of
2264:
2013:
2001:
183:. Often an adjective is added for specifying the operation, such as in
2449:, since these two numbers are the only units in the ring of integers.
4638:
The left inverse can be used to determine the least norm solution of
3610:
3582:
the only element with a two-sided inverse is the identity element 1.
2098:
1795:
5828:, p. 15 (def in unital magma) and p. 33 (def in semigroup)
2804:
A function has a left inverse or a right inverse if and only it is
3416:
every element has a unique two-sided inverse (namely itself), but
3314:
A unital magma in which all elements are invertible need not be a
1666:
3865:
acts a right identity, and the left/right roles are reversed for
2437:
for distinguishing it from matrices that are invertible over the
1194:
under an operation if it has a left inverse and a right inverse.
928:. This lack of inverses is the main motivation for extending the
1068:{\textstyle n\mapsto \left\lfloor {\frac {n}{2}}\right\rfloor ,}
4740:{\displaystyle x=\left(A^{\text{T}}A\right)^{-1}A^{\text{T}}b.}
2828:. This terminology is also used for morphisms in any category.
2371:, that is, an identity element for matrix multiplication is a
1295:
of an invertible element is its unique left or right inverse.
655:
for which the left-hand sides of the equalities are defined.
4148:
All examples in this section involve associative operators.
2661:
In the function and homomorphism cases, this means that the
1298:
If the operation is denoted as an addition, the inverse, or
4061:; in other words every element has commuting pseudoinverse
3785:
as defined in this section. Another easy to prove fact: if
5873:
Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups,
2121:(for avoiding confusion with additive inverses) of a unit
5859:, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
4081:* is not the pseudoinverse. Rather, the pseudoinverse of
3205:. An element with an inverse element only on one side is
2243:(that is, their product with a nonzero element cannot be
2056:, which are denoted as the usual operations on numbers.
2488:, and share many properties with function composition.
2375:(same number for rows and columns) whose entries of the
3311:
the elements 2 and 3 each have two two-sided inverses.
1205:
are respectively a left inverse and a right inverse of
360:). However, these concepts are also commonly used with
5637:
5588:
5540:
5419:
5364:
5316:
5210:
5152:
5071:
5020:
4974:
4789:
2164:
1482:
1379:
1116:
1089:
1037:
300:
271:
27:
Generalization of additive and multiplicative inverses
5518:
4944:
4857:
4765:
4681:
4644:
4534:
4490:
4381:
4337:
4306:
4282:
4252:
4224:
4201:
3765:. Every regular element has at least one inverse: if
3647:
3627:
3591:
3422:
3324:
3222:
3191:
3167:
3147:
3119:
3099:
3079:
3059:
3035:
3015:
2991:
2959:
2927:
2901:
2878:
2847:
2737:
2703:
2671:
2633:
2602:
2576:
2536:
2504:
2131:
1974:
1941:
1912:
1880:
1763:
1740:
1720:
1690:
1576:
1550:
1524:
1445:
1409:
1342:
1312:
1218:
1008:
985:
948:
848:
779:
727:
698:
672:
596:
548:
446:
409:
298:
269:
2763:{\displaystyle \operatorname {id} _{X}\colon X\to X}
2255:
from the ring of integers, and, more generally, the
3869:. This simple observation can be generalized using
5697:
5498:
4927:
4832:
4739:
4659:
4627:
4520:
4474:
4367:
4312:
4288:
4258:
4230:
4207:
3662:
3633:
3597:
3440:
3342:
3240:
3197:
3173:
3153:
3125:
3105:
3085:
3065:
3041:
3021:
2997:
2977:
2945:
2913:
2884:
2853:
2762:
2709:
2677:
2653:
2619:
2588:
2562:
2522:
2398:is invertible if and only if its determinant is a
2286:, and straightforwardly extended to matrices over
2180:
2150:
1986:
1959:
1927:
1898:
1850:. In this monoid, the invertible elements are the
1772:
1749:
1726:
1706:
1608:
1562:
1536:
1495:
1464:
1431:
1395:
1361:
1324:
1281:
1163:, and it has a right inverse if and only if it is
1155:More generally, a function has a left inverse for
1140:
1102:
1067:
1023:
994:
963:
869:
791:
754:
721:(This results immediately from the definition, by
713:
684:
637:{\displaystyle x*e=x\quad {\text{and}}\quad e*y=y}
636:
554:
494:
418:
313:
284:
3877:in an arbitrary semigroup is a left identity for
526:Examples of non-total associative operations are
30:"Invertible" redirects here. For other uses, see
3448:is not a loop because the Cayley table is not a
2789:A function is invertible if and only if it is a
4069:. In contrast, a subclass of *-semigroups, the
1854:; the elements that have left inverses are the
124:. (An identity element is an element such that
4246:is different from zero. If the determinant of
3093:is both a left inverse and a right inverse of
2251:, which produces, in particular, the field of
332:, except when otherwise stated and in section
4156:The lower and upper adjoints in a (monotone)
2194:is never a unit, except when the ring is the
1858:, and those that have right inverses are the
8:
4756:As an example of matrix inverses, consider:
3923:0 because 000 = 0, whereas a group may not.
2693:. In the morphism case, this means that the
2627:or, in the function and homomorphism cases,
2158:or, when the multiplication is commutative,
380:. It follows that the common definitions of
195:. In this case (associative operation), an
160:for which the left-hand sides are defined.)
4168:are quasi-inverses of each other; that is,
2213:if the multiplication is commutative, or a
222:—where invertible elements are also called
4049:-semigroups in which one additionally has
5632:
5583:
5535:
5523:
5517:
5483:
5478:
5414:
5404:
5359:
5311:
5301:
5285:
5274:
5255:
5205:
5195:
5183:
5147:
5130:
5119:
5066:
5015:
4969:
4956:
4945:
4943:
4913:
4902:
4883:
4867:
4862:
4856:
4784:
4764:
4725:
4712:
4698:
4680:
4643:
4619:
4601:
4596:
4591:
4576:
4565:
4546:
4539:
4533:
4489:
4466:
4445:
4440:
4435:
4423:
4410:
4396:
4383:
4380:
4336:
4305:
4281:
4251:
4223:
4200:
3646:
3626:
3590:
3421:
3323:
3221:
3190:
3166:
3161:. An element with a two-sided inverse in
3146:
3118:
3098:
3078:
3058:
3034:
3014:
2990:
2958:
2926:
2900:
2877:
2846:
2742:
2736:
2702:
2670:
2632:
2601:
2575:
2535:
2503:
2165:
2163:
2136:
2130:
1996:This extension of a monoid is allowed by
1973:
1940:
1911:
1879:
1762:
1739:
1719:
1695:
1689:
1594:
1581:
1575:
1549:
1523:
1483:
1481:
1450:
1444:
1414:
1408:
1380:
1378:
1347:
1341:
1311:
1217:
1117:
1115:
1090:
1088:
1048:
1036:
1007:
984:
947:
847:
778:
726:
697:
671:
613:
595:
547:
445:
408:
299:
297:
270:
268:
5870:, 24(1), December 1982, pp. 173–187
5509:The left inverse doesn't exist, because
3457:
3355:
3250:
971:has infinitely many left inverses under
218:—where every element is invertible, and
199:is an element that has an inverse. In a
5748:
4273:More generally, a square matrix over a
2306:, because of the use of the concept of
1846:from a set to itself is a monoid under
1684:of this set. In this case, the inverse
939:from the integers to the integers. The
4521:{\displaystyle A:m\times n\mid m<n}
4368:{\displaystyle A:m\times n\mid m>n}
4077:. In this case however the involution
2457:Functions, homomorphisms and morphisms
2300:in this section, only matrices over a
881:is an identity element, one says that
4528:we have right inverses; for example,
4013:, in which the interaction axiom is (
2484:into operations that are also called
7:
4375:we have left inverses; for example,
3994:, in which the interaction axiom is
2209:is the only non-unit, the ring is a
1282:{\displaystyle l=l*(x*r)=(l*x)*r=r.}
666:are two identity elements such that
333:
324:In this article, the operations are
5816:M. Kilp, U. Knauer, A.V. Mikhalev,
2797:. In the other cases, one talks of
1031:is a right inverse of the function
249:The word 'inverse' is derived from
5857:Regular semigroups with involution
4025:°. Such an operation is called an
2097:Under multiplication, a ring is a
1570:is invertible, and its inverse is
25:
4327:have several one-sided inverses:
4300:its determinant is invertible in
4188:Generalized inverses of matrices
2230:infinite-dimensional vector space
2071:; it has an identity, called the
1874:has the cancellation property if
1174:, right inverses are also called
5834:Fundamentals of Semigroup Theory
4935:By components it is computed as
2563:{\displaystyle g\colon Y'\to Z,}
2491:In all the case, composition is
924:is the only element that has an
340:Definitions and basic properties
5864:P-systems in regular semigroups
3984:-semigroups have been studied:
3773:then it is easy to verify that
2331:matrix (that is, a matrix with
2113:for multiplication is called a
2063:, which means that addition is
1178:, and left inverses are called
773:are different identities, then
618:
612:
566:associative operation on a set
495:{\displaystyle x*(y*z)=(x*y)*z}
314:{\displaystyle {\tfrac {y}{x}}}
285:{\displaystyle {\tfrac {x}{y}}}
234:. This has been generalized to
3657:
3651:
3435:
3423:
3337:
3325:
3235:
3223:
2754:
2551:
2523:{\displaystyle f\colon X\to Y}
2514:
1261:
1249:
1243:
1231:
1141:{\textstyle {\frac {n-1}{2}},}
1041:
952:
483:
471:
465:
453:
214:Inverses are commonly used in
1:
4043:completely regular semigroups
3711:if there exists some element
2685:equals or is included in the
2059:Under addition, a ring is an
1609:{\displaystyle y^{-1}x^{-1}.}
4067:completely simple semigroups
3972:. Although it may seem that
3318:. For example, in the magma
2654:{\displaystyle Y\subset Y'.}
2383:, and all other entries are
2367:, and only in this case. An
2181:{\textstyle {\frac {1}{x}}.}
1842:For example, the set of the
1465:{\displaystyle f^{\circ -1}}
1396:{\textstyle {\frac {1}{x}}.}
802:For example, in the case of
45:generalises the concepts of
4851:, we have a right inverse,
4037:Clearly a group is both an
4029:, and typically denoted by
3857:acts as a left identity on
3216:Elements of a unital magma
2782:), which is called also an
2406:(that is, is invertible in
2083:has an inverse, called its
1496:{\textstyle {\frac {1}{f}}}
1103:{\textstyle {\frac {n}{2}}}
964:{\displaystyle x\mapsto 2x}
819:for every positive integer
238:, where, by definition, an
32:Invertible (disambiguation)
5917:
5712:, and cannot be inverted.
4136:
3684:
2596:is defined if and only if
2247:). This is the process of
1332:Otherwise, the inverse of
830:are identity elements for
528:multiplication of matrices
323:
29:
5780:Howie, prop. 2.3.3, p. 51
5767:as identity elements for
5759:as identity elements for
3976:° will be the inverse of
2778:, algebraic structure or
2016:and, more generally, the
1472:is not commonly used for
352:are commonly defined for
3475:
3472:
3469:
3466:
3463:
3460:
3367:
3364:
3361:
3358:
3262:
3259:
3256:
3253:
2946:{\displaystyle a,b\in S}
2589:{\displaystyle g\circ f}
2278:is commonly defined for
1817:under monoid operation.
1432:{\displaystyle x^{*-1}.}
755:{\displaystyle e=e*f=f.}
5832:Howie, John M. (1995).
4323:Non-square matrices of
3884:and right identity for
2202:as its unique element.
2151:{\displaystyle x^{-1},}
2103:multiplicative identity
1661:Thus, the inverse is a
1362:{\displaystyle x^{-1},}
838:Left and right inverses
530:of arbitrary size, and
433:A partial operation is
5699:
5500:
4929:
4834:
4741:
4661:
4629:
4522:
4476:
4369:
4314:
4290:
4260:
4232:
4209:
4085:is the unique element
3741:is called (simply) an
3731:is sometimes called a
3664:
3635:
3599:
3442:
3344:
3242:
3199:
3175:
3155:
3127:
3107:
3087:
3067:
3043:
3023:
2999:
2979:
2947:
2915:
2914:{\displaystyle e\in S}
2886:
2855:
2786:in the function case.
2764:
2711:
2679:
2655:
2621:
2590:
2564:
2524:
2190:The additive identity
2182:
2152:
2119:multiplicative inverse
1988:
1961:
1929:
1900:
1774:
1751:
1728:
1708:
1707:{\displaystyle g^{-1}}
1610:
1564:
1538:
1505:multiplicative inverse
1497:
1466:
1433:
1397:
1363:
1326:
1283:
1142:
1104:
1069:
1025:
996:
965:
871:
870:{\displaystyle x*y=e,}
793:
756:
715:
686:
638:
556:
496:
420:
334:§ Generalizations
315:
286:
254:
189:multiplicative inverse
5769:matrix multiplication
5727:Latin square property
5700:
5501:
4930:
4835:
4742:
4662:
4630:
4523:
4477:
4370:
4315:
4291:
4261:
4242:) if and only if its
4240:matrix multiplication
4233:
4210:
4127:Moore–Penrose inverse
4075:Moore–Penrose inverse
3709:(von Neumann) regular
3665:
3636:
3600:
3443:
3441:{\displaystyle (S,*)}
3345:
3343:{\displaystyle (S,*)}
3243:
3241:{\displaystyle (S,*)}
3200:
3176:
3156:
3128:
3108:
3088:
3068:
3044:
3024:
3000:
2980:
2978:{\displaystyle a*b=e}
2948:
2916:
2887:
2856:
2765:
2712:
2680:
2656:
2622:
2591:
2565:
2525:
2414:can be computed with
2276:Matrix multiplication
2183:
2153:
2048:with two operations,
1989:
1962:
1960:{\displaystyle yx=zx}
1930:
1901:
1899:{\displaystyle xy=xz}
1868:cancellation property
1800:associative operation
1775:
1752:
1729:
1709:
1656:associative operation
1611:
1565:
1539:
1498:
1467:
1434:
1398:
1369:or, in the case of a
1364:
1336:is generally denoted
1327:
1284:
1159:if and only if it is
1143:
1105:
1070:
1026:
997:
966:
872:
804:matrix multiplication
794:
757:
716:
687:
639:
557:
497:
421:
366:matrix multiplication
316:
287:
18:Inverse (ring theory)
5877:, 16(1978), 369–377.
5761:function composition
5516:
4942:
4855:
4763:
4679:
4667:, which is also the
4660:{\displaystyle Ax=b}
4642:
4532:
4488:
4379:
4335:
4304:
4280:
4250:
4222:
4199:
4139:Quasiregular element
4071:*-regular semigroups
3663:{\displaystyle U(S)}
3645:
3625:
3589:
3420:
3322:
3220:
3189:
3165:
3145:
3117:
3097:
3077:
3057:
3033:
3013:
2989:
2957:
2925:
2899:
2876:
2845:
2799:inverse isomorphisms
2735:
2701:
2669:
2631:
2620:{\displaystyle Y'=Y}
2600:
2574:
2534:
2502:
2474:algebraic structures
2468:that generalizes to
2410:. In this case, its
2353:matrix, the product
2162:
2129:
2079:; and every element
1972:
1939:
1928:{\displaystyle y=z,}
1910:
1878:
1860:surjective functions
1848:function composition
1761:
1738:
1718:
1688:
1574:
1548:
1522:
1518:are invertible, and
1503:can be used for the
1480:
1474:function composition
1443:
1407:
1377:
1340:
1310:
1216:
1157:function composition
1114:
1087:
1035:
1024:{\displaystyle 2n+1}
1006:
983:
973:function composition
946:
910:nonnegative integers
846:
832:function composition
777:
725:
714:{\displaystyle e=f.}
696:
670:
594:
546:
532:function composition
444:
407:
370:function composition
296:
267:
41:, the concept of an
5491:
4875:
4609:
4453:
4123:generalized inverse
3932:Generalized inverse
3873:: every idempotent
2222:noncommutative ring
2111:invertible element
2046:algebraic structure
1987:{\displaystyle y=z}
1856:injective functions
1852:bijective functions
1835:If a monoid is not
1813:in a monoid form a
1811:invertible elements
1714:of a group element
1669:one. It is also an
1563:{\displaystyle x*y}
1537:{\displaystyle x*y}
1325:{\displaystyle -x.}
932:into the integers.
792:{\displaystyle e*f}
685:{\displaystyle e*f}
372:and composition of
163:When the operation
5757:identity functions
5737:Unit (ring theory)
5695:
5689:
5623:
5577:
5496:
5494:
5474:
5465:
5395:
5353:
5241:
5177:
5096:
5057:
5009:
4925:
4858:
4830:
4824:
4737:
4657:
4625:
4611:
4592:
4589:
4518:
4472:
4455:
4436:
4433:
4365:
4310:
4286:
4256:
4228:
4215:with entries in a
4205:
4152:Galois connections
3907:, the elements of
3660:
3631:
3595:
3438:
3340:
3238:
3195:
3171:
3151:
3123:
3103:
3083:
3063:
3039:
3019:
2995:
2975:
2943:
2911:
2882:
2851:
2826:split monomorphism
2760:
2707:
2675:
2651:
2617:
2586:
2560:
2520:
2257:field of fractions
2178:
2148:
2018:field of fractions
1998:Grothendieck group
1984:
1957:
1925:
1896:
1784:Rubik's cube group
1773:{\displaystyle g.}
1770:
1750:{\displaystyle g,}
1747:
1724:
1704:
1634:is also called an
1606:
1560:
1534:
1493:
1462:
1429:
1393:
1359:
1322:
1279:
1148:depending whether
1138:
1100:
1065:
1021:
995:{\displaystyle 2n}
992:
961:
867:
828:identity functions
789:
752:
711:
682:
634:
552:
492:
419:{\displaystyle *.}
416:
362:partial operations
350:invertible element
311:
309:
282:
280:
205:invertible element
197:invertible element
193:functional inverse
5901:Binary operations
5765:identity matrices
5526:
5481:
5412:
5309:
5277:
5258:
5203:
5122:
4959:
4905:
4886:
4865:
4728:
4701:
4599:
4568:
4549:
4540:
4538:
4443:
4426:
4399:
4384:
4382:
4313:{\displaystyle R}
4289:{\displaystyle R}
4268:invertible matrix
4259:{\displaystyle M}
4231:{\displaystyle K}
4208:{\displaystyle M}
4158:Galois connection
3921:absorbing element
3917:inverse semigroup
3871:Green's relations
3789:is an inverse of
3781:is an inverse of
3687:Regular semigroup
3641:, and denoted by
3634:{\displaystyle S}
3598:{\displaystyle *}
3585:If the operation
3580:
3579:
3414:
3413:
3309:
3308:
3198:{\displaystyle S}
3174:{\displaystyle S}
3154:{\displaystyle y}
3135:two-sided inverse
3126:{\displaystyle x}
3106:{\displaystyle y}
3086:{\displaystyle x}
3066:{\displaystyle a}
3042:{\displaystyle b}
3022:{\displaystyle b}
2998:{\displaystyle a}
2885:{\displaystyle *}
2854:{\displaystyle S}
2837:In a unital magma
2822:split epimorphism
2784:identity function
2770:for every object
2710:{\displaystyle f}
2678:{\displaystyle f}
2466:partial operation
2435:unimodular matrix
2392:invertible matrix
2379:are all equal to
2173:
2117:. The inverse or
2073:additive identity
2030:commutative rings
1798:is a set with an
1782:For example, the
1727:{\displaystyle g}
1544:is defined, then
1491:
1388:
1133:
1098:
1056:
941:doubling function
918:additive identity
799:is not defined.
692:is defined, then
616:
555:{\displaystyle *}
538:Identity elements
391:In this section,
354:binary operations
330:identity elements
308:
279:
242:is an invertible
232:inverse functions
16:(Redirected from
5908:
5896:Abstract algebra
5851:
5804:
5803:
5796:
5790:
5787:
5781:
5778:
5772:
5753:
5704:
5702:
5701:
5696:
5694:
5693:
5628:
5627:
5582:
5581:
5528:
5527:
5524:
5505:
5503:
5502:
5497:
5495:
5490:
5482:
5479:
5470:
5469:
5413:
5405:
5400:
5399:
5358:
5357:
5310:
5302:
5293:
5292:
5284:
5280:
5279:
5278:
5275:
5260:
5259:
5256:
5246:
5245:
5204:
5196:
5191:
5190:
5182:
5181:
5138:
5137:
5129:
5125:
5124:
5123:
5120:
5101:
5100:
5062:
5061:
5014:
5013:
4961:
4960:
4957:
4934:
4932:
4931:
4926:
4921:
4920:
4912:
4908:
4907:
4906:
4903:
4888:
4887:
4884:
4874:
4866:
4863:
4839:
4837:
4836:
4831:
4829:
4828:
4746:
4744:
4743:
4738:
4730:
4729:
4726:
4720:
4719:
4711:
4707:
4703:
4702:
4699:
4675:and is given by
4666:
4664:
4663:
4658:
4634:
4632:
4631:
4626:
4624:
4623:
4610:
4608:
4600:
4597:
4590:
4585:
4584:
4583:
4575:
4571:
4570:
4569:
4566:
4551:
4550:
4547:
4527:
4525:
4524:
4519:
4481:
4479:
4478:
4473:
4471:
4470:
4454:
4452:
4444:
4441:
4434:
4429:
4428:
4427:
4424:
4418:
4417:
4409:
4405:
4401:
4400:
4397:
4374:
4372:
4371:
4366:
4319:
4317:
4316:
4311:
4295:
4293:
4292:
4287:
4275:commutative ring
4265:
4263:
4262:
4257:
4237:
4235:
4234:
4229:
4214:
4212:
4211:
4206:
3669:
3667:
3666:
3661:
3640:
3638:
3637:
3632:
3604:
3602:
3601:
3596:
3458:
3447:
3445:
3444:
3439:
3356:
3349:
3347:
3346:
3341:
3251:
3247:
3245:
3244:
3239:
3211:right invertible
3204:
3202:
3201:
3196:
3180:
3178:
3177:
3172:
3160:
3158:
3157:
3152:
3132:
3130:
3129:
3124:
3112:
3110:
3109:
3104:
3092:
3090:
3089:
3084:
3073:. If an element
3072:
3070:
3069:
3064:
3048:
3046:
3045:
3040:
3028:
3026:
3025:
3020:
3004:
3002:
3001:
2996:
2984:
2982:
2981:
2976:
2952:
2950:
2949:
2944:
2920:
2918:
2917:
2912:
2894:identity element
2891:
2889:
2888:
2883:
2871:binary operation
2860:
2858:
2857:
2852:
2795:inverse function
2773:
2769:
2767:
2766:
2761:
2747:
2746:
2724:
2716:
2714:
2713:
2708:
2692:
2684:
2682:
2681:
2676:
2660:
2658:
2657:
2652:
2647:
2626:
2624:
2623:
2618:
2610:
2595:
2593:
2592:
2587:
2570:the composition
2569:
2567:
2566:
2561:
2550:
2529:
2527:
2526:
2521:
2448:
2444:
2431:integer matrices
2424:
2409:
2405:
2397:
2386:
2382:
2366:
2356:
2352:
2342:
2338:
2334:
2330:
2320:
2302:commutative ring
2253:rational numbers
2246:
2237:commutative ring
2226:linear functions
2208:
2201:
2193:
2187:
2185:
2184:
2179:
2174:
2166:
2157:
2155:
2154:
2149:
2144:
2143:
2124:
2108:
2093:
2085:additive inverse
2082:
2078:
2010:rational numbers
1995:
1993:
1991:
1990:
1985:
1966:
1964:
1963:
1958:
1934:
1932:
1931:
1926:
1905:
1903:
1902:
1897:
1873:
1804:identity element
1779:
1777:
1776:
1771:
1756:
1754:
1753:
1748:
1733:
1731:
1730:
1725:
1713:
1711:
1710:
1705:
1703:
1702:
1630:, an invertible
1615:
1613:
1612:
1607:
1602:
1601:
1589:
1588:
1569:
1567:
1566:
1561:
1543:
1541:
1540:
1535:
1517:
1513:
1502:
1500:
1499:
1494:
1492:
1484:
1471:
1469:
1468:
1463:
1461:
1460:
1438:
1436:
1435:
1430:
1425:
1424:
1402:
1400:
1399:
1394:
1389:
1381:
1368:
1366:
1365:
1360:
1355:
1354:
1335:
1331:
1329:
1328:
1323:
1305:
1302:, of an element
1300:additive inverse
1288:
1286:
1285:
1280:
1208:
1204:
1200:
1152:is even or odd.
1151:
1147:
1145:
1144:
1139:
1134:
1129:
1118:
1109:
1107:
1106:
1101:
1099:
1091:
1082:
1074:
1072:
1071:
1066:
1061:
1057:
1049:
1030:
1028:
1027:
1022:
1001:
999:
998:
993:
978:
970:
968:
967:
962:
926:additive inverse
923:
915:
904:
896:
892:
884:
880:
876:
874:
873:
868:
822:
815:
798:
796:
795:
790:
772:
768:
761:
759:
758:
753:
720:
718:
717:
712:
691:
689:
688:
683:
665:
661:
654:
650:
643:
641:
640:
635:
617:
614:
586:
576:identity element
569:
561:
559:
558:
553:
522:
518:
501:
499:
498:
493:
425:
423:
422:
417:
394:
386:identity element
344:The concepts of
320:
318:
317:
312:
310:
301:
291:
289:
288:
283:
281:
272:
228:inverse matrices
207:, also called a
185:additive inverse
174:
171:, if an element
166:
159:
155:
151:
137:
123:
115:
111:
103:
100:, one says that
99:
85:
80:identity element
77:
66:
55:
21:
5916:
5915:
5911:
5910:
5909:
5907:
5906:
5905:
5881:
5880:
5875:Semigroup Forum
5868:Semigroup Forum
5862:Miyuki Yamada,
5848:
5838:Clarendon Press
5831:
5813:
5808:
5807:
5798:
5797:
5793:
5788:
5784:
5779:
5775:
5754:
5750:
5745:
5718:
5710:singular matrix
5688:
5687:
5682:
5677:
5671:
5670:
5665:
5660:
5654:
5653:
5648:
5643:
5633:
5622:
5621:
5616:
5611:
5605:
5604:
5599:
5594:
5584:
5576:
5575:
5570:
5564:
5563:
5558:
5552:
5551:
5546:
5536:
5519:
5514:
5513:
5493:
5492:
5464:
5463:
5455:
5449:
5448:
5443:
5434:
5433:
5428:
5415:
5394:
5393:
5388:
5379:
5378:
5370:
5360:
5352:
5351:
5346:
5340:
5339:
5334:
5328:
5327:
5322:
5312:
5294:
5270:
5266:
5262:
5261:
5251:
5248:
5247:
5240:
5239:
5234:
5225:
5224:
5216:
5206:
5176:
5175:
5170:
5164:
5163:
5158:
5148:
5146:
5139:
5115:
5111:
5107:
5106:
5103:
5102:
5095:
5094:
5089:
5083:
5082:
5077:
5067:
5056:
5055:
5050:
5044:
5043:
5038:
5032:
5031:
5026:
5016:
5008:
5007:
5002:
4997:
4991:
4990:
4985:
4980:
4970:
4962:
4952:
4940:
4939:
4898:
4894:
4890:
4889:
4879:
4853:
4852:
4823:
4822:
4817:
4812:
4806:
4805:
4800:
4795:
4785:
4761:
4760:
4721:
4694:
4693:
4689:
4688:
4677:
4676:
4640:
4639:
4615:
4561:
4557:
4553:
4552:
4542:
4541:
4530:
4529:
4486:
4485:
4462:
4419:
4392:
4391:
4387:
4386:
4385:
4377:
4376:
4333:
4332:
4302:
4301:
4278:
4277:
4248:
4247:
4220:
4219:
4197:
4196:
4190:
4154:
4146:
4141:
4135:
3943:
3913:
3901:
3889:
3882:
3699:In a semigroup
3689:
3683:
3676:
3643:
3642:
3623:
3622:
3587:
3586:
3418:
3417:
3320:
3319:
3218:
3217:
3207:left invertible
3187:
3186:
3163:
3162:
3143:
3142:
3137:, or simply an
3115:
3114:
3095:
3094:
3075:
3074:
3055:
3054:
3031:
3030:
3011:
3010:
2987:
2986:
2955:
2954:
2923:
2922:
2897:
2896:
2874:
2873:
2843:
2842:
2839:
2834:
2832:Generalizations
2771:
2738:
2733:
2732:
2722:
2699:
2698:
2690:
2667:
2666:
2640:
2629:
2628:
2603:
2598:
2597:
2572:
2571:
2543:
2532:
2531:
2500:
2499:
2459:
2446:
2442:
2429:In the case of
2422:
2407:
2403:
2395:
2384:
2380:
2369:identity matrix
2358:
2354:
2344:
2340:
2336:
2332:
2322:
2318:
2273:
2261:integral domain
2244:
2206:
2199:
2191:
2160:
2159:
2132:
2127:
2126:
2122:
2106:
2088:
2080:
2076:
2038:
2022:integral domain
2006:natural numbers
1970:
1969:
1968:
1937:
1936:
1908:
1907:
1876:
1875:
1871:
1792:
1759:
1758:
1736:
1735:
1716:
1715:
1691:
1686:
1685:
1682:transformations
1644:
1628:category theory
1590:
1577:
1572:
1571:
1546:
1545:
1520:
1519:
1515:
1511:
1478:
1477:
1446:
1441:
1440:
1410:
1405:
1404:
1375:
1374:
1373:multiplication
1343:
1338:
1337:
1333:
1308:
1307:
1303:
1214:
1213:
1206:
1202:
1198:
1188:
1172:category theory
1149:
1119:
1112:
1111:
1085:
1084:
1080:
1044:
1033:
1032:
1004:
1003:
981:
980:
976:
944:
943:
930:natural numbers
921:
913:
902:
894:
890:
882:
878:
844:
843:
840:
820:
817:identity matrix
807:
806:, there is one
775:
774:
770:
766:
723:
722:
694:
693:
668:
667:
663:
659:
652:
648:
592:
591:
584:
579:, or simply an
567:
544:
543:
540:
520:
506:
442:
441:
431:
405:
404:
392:
346:inverse element
342:
337:
294:
293:
265:
264:
236:category theory
177:inverse element
172:
164:
157:
153:
139:
125:
121:
113:
109:
101:
87:
83:
75:
67:) of numbers.
61:
50:
43:inverse element
35:
28:
23:
22:
15:
12:
11:
5:
5914:
5912:
5904:
5903:
5898:
5893:
5883:
5882:
5879:
5878:
5871:
5860:
5855:Drazin, M.P.,
5853:
5846:
5829:
5812:
5809:
5806:
5805:
5791:
5782:
5773:
5747:
5746:
5744:
5741:
5740:
5739:
5734:
5732:Loop (algebra)
5729:
5724:
5717:
5714:
5706:
5705:
5692:
5686:
5683:
5681:
5678:
5676:
5673:
5672:
5669:
5666:
5664:
5661:
5659:
5656:
5655:
5652:
5649:
5647:
5644:
5642:
5639:
5638:
5636:
5631:
5626:
5620:
5617:
5615:
5612:
5610:
5607:
5606:
5603:
5600:
5598:
5595:
5593:
5590:
5589:
5587:
5580:
5574:
5571:
5569:
5566:
5565:
5562:
5559:
5557:
5554:
5553:
5550:
5547:
5545:
5542:
5541:
5539:
5534:
5531:
5522:
5507:
5506:
5489:
5486:
5477:
5473:
5468:
5462:
5459:
5456:
5454:
5451:
5450:
5447:
5444:
5442:
5439:
5436:
5435:
5432:
5429:
5427:
5424:
5421:
5420:
5418:
5411:
5408:
5403:
5398:
5392:
5389:
5387:
5384:
5381:
5380:
5377:
5374:
5371:
5369:
5366:
5365:
5363:
5356:
5350:
5347:
5345:
5342:
5341:
5338:
5335:
5333:
5330:
5329:
5326:
5323:
5321:
5318:
5317:
5315:
5308:
5305:
5300:
5297:
5295:
5291:
5288:
5283:
5273:
5269:
5265:
5254:
5250:
5249:
5244:
5238:
5235:
5233:
5230:
5227:
5226:
5223:
5220:
5217:
5215:
5212:
5211:
5209:
5202:
5199:
5194:
5189:
5186:
5180:
5174:
5171:
5169:
5166:
5165:
5162:
5159:
5157:
5154:
5153:
5151:
5145:
5142:
5140:
5136:
5133:
5128:
5118:
5114:
5110:
5105:
5104:
5099:
5093:
5090:
5088:
5085:
5084:
5081:
5078:
5076:
5073:
5072:
5070:
5065:
5060:
5054:
5051:
5049:
5046:
5045:
5042:
5039:
5037:
5034:
5033:
5030:
5027:
5025:
5022:
5021:
5019:
5012:
5006:
5003:
5001:
4998:
4996:
4993:
4992:
4989:
4986:
4984:
4981:
4979:
4976:
4975:
4973:
4968:
4965:
4963:
4955:
4951:
4948:
4947:
4924:
4919:
4916:
4911:
4901:
4897:
4893:
4882:
4878:
4873:
4870:
4861:
4841:
4840:
4827:
4821:
4818:
4816:
4813:
4811:
4808:
4807:
4804:
4801:
4799:
4796:
4794:
4791:
4790:
4788:
4783:
4780:
4777:
4774:
4771:
4768:
4751:rank deficient
4736:
4733:
4724:
4718:
4715:
4710:
4706:
4697:
4692:
4687:
4684:
4656:
4653:
4650:
4647:
4636:
4635:
4622:
4618:
4614:
4607:
4604:
4595:
4588:
4582:
4579:
4574:
4564:
4560:
4556:
4545:
4537:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4482:
4469:
4465:
4461:
4458:
4451:
4448:
4439:
4432:
4422:
4416:
4413:
4408:
4404:
4395:
4390:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4309:
4298:if and only if
4296:is invertible
4285:
4255:
4227:
4204:
4189:
4186:
4153:
4150:
4145:
4142:
4137:Main article:
4134:
4131:
4035:
4034:
4006:
3961:; this endows
3942:
3936:
3911:
3899:
3887:
3880:
3685:Main article:
3682:
3681:In a semigroup
3679:
3674:
3659:
3656:
3653:
3650:
3630:
3619:group of units
3594:
3578:
3577:
3574:
3571:
3568:
3565:
3562:
3558:
3557:
3554:
3551:
3548:
3545:
3542:
3538:
3537:
3534:
3531:
3528:
3525:
3522:
3518:
3517:
3514:
3511:
3508:
3505:
3502:
3498:
3497:
3494:
3491:
3488:
3485:
3482:
3478:
3477:
3474:
3471:
3468:
3465:
3462:
3437:
3434:
3431:
3428:
3425:
3412:
3411:
3408:
3405:
3402:
3398:
3397:
3394:
3391:
3388:
3384:
3383:
3380:
3377:
3374:
3370:
3369:
3366:
3363:
3360:
3339:
3336:
3333:
3330:
3327:
3307:
3306:
3303:
3300:
3297:
3293:
3292:
3289:
3286:
3283:
3279:
3278:
3275:
3272:
3269:
3265:
3264:
3261:
3258:
3255:
3237:
3234:
3231:
3228:
3225:
3194:
3170:
3150:
3122:
3102:
3082:
3062:
3038:
3018:
2994:
2974:
2971:
2968:
2965:
2962:
2942:
2939:
2936:
2933:
2930:
2910:
2907:
2904:
2881:
2850:
2838:
2835:
2833:
2830:
2759:
2756:
2753:
2750:
2745:
2741:
2706:
2674:
2650:
2646:
2643:
2639:
2636:
2616:
2613:
2609:
2606:
2585:
2582:
2579:
2559:
2556:
2553:
2549:
2546:
2542:
2539:
2519:
2516:
2513:
2510:
2507:
2458:
2455:
2412:inverse matrix
2357:is defined if
2339:columns), and
2304:are considered
2272:
2269:
2177:
2172:
2169:
2147:
2142:
2139:
2135:
2075:, and denoted
2054:multiplication
2037:
2034:
1983:
1980:
1977:
1956:
1953:
1950:
1947:
1944:
1924:
1921:
1918:
1915:
1895:
1892:
1889:
1886:
1883:
1830:group of units
1791:
1788:
1769:
1766:
1746:
1743:
1723:
1701:
1698:
1694:
1643:
1640:
1618:An invertible
1605:
1600:
1597:
1593:
1587:
1584:
1580:
1559:
1556:
1553:
1533:
1530:
1527:
1490:
1487:
1459:
1456:
1453:
1449:
1428:
1423:
1420:
1417:
1413:
1392:
1387:
1384:
1358:
1353:
1350:
1346:
1321:
1318:
1315:
1290:
1289:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1190:An element is
1187:
1184:
1137:
1132:
1128:
1125:
1122:
1097:
1094:
1077:floor function
1064:
1060:
1055:
1052:
1047:
1043:
1040:
1020:
1017:
1014:
1011:
991:
988:
960:
957:
954:
951:
866:
863:
860:
857:
854:
851:
839:
836:
788:
785:
782:
751:
748:
745:
742:
739:
736:
733:
730:
710:
707:
704:
701:
681:
678:
675:
645:
644:
633:
630:
627:
624:
621:
611:
608:
605:
602:
599:
583:is an element
562:be a possibly
551:
539:
536:
503:
502:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
430:
427:
415:
412:
341:
338:
307:
304:
278:
275:
179:or simply the
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5913:
5902:
5899:
5897:
5894:
5892:
5889:
5888:
5886:
5876:
5872:
5869:
5865:
5861:
5858:
5854:
5849:
5847:0-19-851194-9
5843:
5839:
5835:
5830:
5827:
5826:3-11-015248-7
5823:
5819:
5815:
5814:
5810:
5801:
5795:
5792:
5786:
5783:
5777:
5774:
5770:
5766:
5762:
5758:
5752:
5749:
5742:
5738:
5735:
5733:
5730:
5728:
5725:
5723:
5722:Division ring
5720:
5719:
5715:
5713:
5711:
5690:
5684:
5679:
5674:
5667:
5662:
5657:
5650:
5645:
5640:
5634:
5629:
5624:
5618:
5613:
5608:
5601:
5596:
5591:
5585:
5578:
5572:
5567:
5560:
5555:
5548:
5543:
5537:
5532:
5529:
5520:
5512:
5511:
5510:
5487:
5484:
5475:
5471:
5466:
5460:
5457:
5452:
5445:
5440:
5437:
5430:
5425:
5422:
5416:
5409:
5406:
5401:
5396:
5390:
5385:
5382:
5375:
5372:
5367:
5361:
5354:
5348:
5343:
5336:
5331:
5324:
5319:
5313:
5306:
5303:
5298:
5296:
5289:
5286:
5281:
5271:
5267:
5263:
5252:
5242:
5236:
5231:
5228:
5221:
5218:
5213:
5207:
5200:
5197:
5192:
5187:
5184:
5178:
5172:
5167:
5160:
5155:
5149:
5143:
5141:
5134:
5131:
5126:
5116:
5112:
5108:
5097:
5091:
5086:
5079:
5074:
5068:
5063:
5058:
5052:
5047:
5040:
5035:
5028:
5023:
5017:
5010:
5004:
4999:
4994:
4987:
4982:
4977:
4971:
4966:
4964:
4953:
4949:
4938:
4937:
4936:
4922:
4917:
4914:
4909:
4899:
4895:
4891:
4880:
4876:
4871:
4868:
4859:
4850:
4846:
4825:
4819:
4814:
4809:
4802:
4797:
4792:
4786:
4781:
4778:
4775:
4772:
4769:
4766:
4759:
4758:
4757:
4754:
4752:
4747:
4734:
4731:
4722:
4716:
4713:
4708:
4704:
4695:
4690:
4685:
4682:
4674:
4670:
4669:least squares
4654:
4651:
4648:
4645:
4620:
4616:
4612:
4605:
4602:
4593:
4586:
4580:
4577:
4572:
4562:
4558:
4554:
4543:
4535:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4483:
4467:
4463:
4459:
4456:
4449:
4446:
4437:
4430:
4420:
4414:
4411:
4406:
4402:
4393:
4388:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4330:
4329:
4328:
4326:
4321:
4307:
4299:
4283:
4276:
4271:
4269:
4253:
4245:
4241:
4225:
4218:
4202:
4195:
4194:square matrix
4187:
4185:
4183:
4179:
4175:
4171:
4167:
4163:
4159:
4151:
4149:
4143:
4140:
4132:
4130:
4128:
4124:
4120:
4116:
4112:
4108:
4104:
4100:
4096:
4092:
4088:
4084:
4080:
4076:
4072:
4068:
4064:
4060:
4056:
4052:
4048:
4044:
4040:
4032:
4028:
4024:
4020:
4016:
4012:
4011:
4007:
4005:
4001:
3997:
3993:
3991:
3987:
3986:
3985:
3983:
3979:
3975:
3971:
3969:
3964:
3960:
3956:
3952:
3948:
3940:
3937:
3935:
3933:
3929:
3928:quasi-inverse
3924:
3922:
3918:
3910:
3906:
3902:
3898:
3892:
3890:
3883:
3876:
3872:
3868:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3788:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3748:
3744:
3740:
3737:. An element
3736:
3735:
3734:pseudoinverse
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3697:
3695:
3688:
3680:
3678:
3673:
3654:
3648:
3628:
3620:
3617:, called the
3616:
3612:
3608:
3592:
3583:
3575:
3572:
3569:
3566:
3563:
3560:
3559:
3555:
3552:
3549:
3546:
3543:
3540:
3539:
3535:
3532:
3529:
3526:
3523:
3520:
3519:
3515:
3512:
3509:
3506:
3503:
3500:
3499:
3495:
3492:
3489:
3486:
3483:
3480:
3479:
3459:
3456:
3453:
3451:
3432:
3429:
3426:
3409:
3406:
3403:
3400:
3399:
3395:
3392:
3389:
3386:
3385:
3381:
3378:
3375:
3372:
3371:
3357:
3354:
3353:
3350:given by the
3334:
3331:
3328:
3317:
3312:
3304:
3301:
3298:
3295:
3294:
3290:
3287:
3284:
3281:
3280:
3276:
3273:
3270:
3267:
3266:
3252:
3249:
3232:
3229:
3226:
3214:
3212:
3208:
3192:
3184:
3168:
3148:
3140:
3136:
3120:
3100:
3080:
3060:
3052:
3051:right inverse
3036:
3016:
3008:
2992:
2972:
2969:
2966:
2963:
2960:
2940:
2937:
2934:
2931:
2928:
2908:
2905:
2902:
2895:
2879:
2872:
2868:
2865:, that is, a
2864:
2848:
2836:
2831:
2829:
2827:
2823:
2819:
2815:
2814:vector spaces
2811:
2807:
2802:
2800:
2796:
2792:
2787:
2785:
2781:
2777:
2757:
2751:
2748:
2743:
2739:
2731:
2726:
2720:
2704:
2696:
2688:
2672:
2664:
2648:
2644:
2641:
2637:
2634:
2614:
2611:
2607:
2604:
2583:
2580:
2577:
2557:
2554:
2547:
2544:
2540:
2537:
2517:
2511:
2508:
2505:
2496:
2494:
2489:
2487:
2483:
2479:
2475:
2471:
2470:homomorphisms
2467:
2463:
2456:
2454:
2450:
2440:
2436:
2432:
2427:
2419:
2417:
2416:Cramer's rule
2413:
2401:
2393:
2388:
2378:
2377:main diagonal
2374:
2373:square matrix
2370:
2365:
2361:
2351:
2347:
2329:
2325:
2315:
2313:
2309:
2305:
2303:
2297:
2293:
2289:
2285:
2281:
2277:
2270:
2268:
2266:
2262:
2258:
2254:
2250:
2242:
2241:zero divisors
2238:
2233:
2231:
2227:
2223:
2218:
2216:
2215:division ring
2212:
2203:
2197:
2188:
2175:
2170:
2167:
2145:
2140:
2137:
2133:
2120:
2116:
2112:
2104:
2100:
2095:
2092:
2087:and denoted
2086:
2074:
2070:
2066:
2062:
2061:abelian group
2057:
2055:
2051:
2047:
2043:
2035:
2033:
2031:
2027:
2026:localizations
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1981:
1978:
1975:
1954:
1951:
1948:
1945:
1942:
1922:
1919:
1916:
1913:
1893:
1890:
1887:
1884:
1881:
1869:
1863:
1861:
1857:
1853:
1849:
1845:
1840:
1838:
1833:
1832:of the ring.
1831:
1828:and form the
1827:
1823:
1818:
1816:
1812:
1807:
1805:
1801:
1797:
1789:
1787:
1785:
1780:
1767:
1764:
1744:
1741:
1721:
1699:
1696:
1692:
1683:
1679:
1674:
1672:
1668:
1664:
1659:
1657:
1653:
1649:
1641:
1639:
1637:
1633:
1629:
1625:
1622:is called an
1621:
1616:
1603:
1598:
1595:
1591:
1585:
1582:
1578:
1557:
1554:
1551:
1531:
1528:
1525:
1508:
1506:
1488:
1485:
1475:
1457:
1454:
1451:
1447:
1439:The notation
1426:
1421:
1418:
1415:
1411:
1390:
1385:
1382:
1372:
1356:
1351:
1348:
1344:
1319:
1316:
1313:
1301:
1296:
1294:
1276:
1273:
1270:
1267:
1264:
1258:
1255:
1252:
1246:
1240:
1237:
1234:
1228:
1225:
1222:
1219:
1212:
1211:
1210:
1195:
1193:
1185:
1183:
1181:
1177:
1173:
1168:
1166:
1162:
1158:
1153:
1135:
1130:
1126:
1123:
1120:
1095:
1092:
1078:
1062:
1058:
1053:
1050:
1045:
1038:
1018:
1015:
1012:
1009:
989:
986:
974:
958:
955:
949:
942:
938:
933:
931:
927:
919:
911:
906:
900:
899:right inverse
888:
864:
861:
858:
855:
852:
849:
837:
835:
833:
829:
824:
818:
814:
810:
805:
800:
786:
783:
780:
763:
749:
746:
743:
740:
737:
734:
731:
728:
708:
705:
702:
699:
679:
676:
673:
656:
631:
628:
625:
622:
619:
609:
606:
603:
600:
597:
590:
589:
588:
582:
578:
577:
571:
565:
549:
537:
535:
533:
529:
524:
517:
513:
509:
489:
486:
480:
477:
474:
468:
462:
459:
456:
450:
447:
440:
439:
438:
436:
429:Associativity
428:
426:
413:
410:
402:
398:
389:
387:
383:
382:associativity
379:
375:
371:
367:
363:
359:
355:
351:
347:
339:
335:
331:
327:
322:
305:
302:
276:
273:
262:
258:
257:
252:
247:
245:
241:
237:
233:
229:
225:
221:
217:
212:
210:
206:
202:
198:
194:
190:
186:
182:
178:
170:
161:
150:
146:
142:
136:
132:
128:
119:
118:right inverse
107:
98:
94:
90:
81:
74:denoted here
73:
68:
65:
59:
54:
48:
44:
40:
33:
19:
5863:
5856:
5833:
5817:
5794:
5789:Howie p. 102
5785:
5776:
5751:
5707:
5508:
4848:
4844:
4842:
4755:
4748:
4671:formula for
4637:
4322:
4272:
4191:
4181:
4177:
4173:
4169:
4165:
4161:
4155:
4147:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4078:
4062:
4058:
4054:
4050:
4046:
4045:; these are
4038:
4036:
4030:
4022:
4018:
4014:
4010:*-semigroups
4008:
4003:
3999:
3995:
3989:
3988:
3981:
3977:
3973:
3967:
3966:
3962:
3958:
3954:
3950:
3946:
3944:
3938:
3927:
3925:
3908:
3904:
3896:
3893:
3885:
3878:
3874:
3866:
3862:
3858:
3854:
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3814:
3806:
3802:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3770:
3766:
3762:
3758:
3754:
3750:
3746:
3742:
3738:
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3698:
3690:
3671:
3584:
3581:
3454:
3450:Latin square
3415:
3352:Cayley table
3313:
3310:
3215:
3210:
3206:
3182:
3138:
3134:
3133:is called a
3050:
3049:is called a
3007:left inverse
3006:
3005:is called a
2861:be a unital
2840:
2816:but not for
2803:
2798:
2788:
2729:
2728:There is an
2727:
2497:
2490:
2485:
2460:
2451:
2439:real numbers
2428:
2420:
2389:
2363:
2359:
2349:
2345:
2327:
2323:
2316:
2299:
2274:
2249:localization
2234:
2219:
2204:
2198:, which has
2189:
2110:
2105:and denoted
2096:
2090:
2058:
2053:
2049:
2039:
1870:(an element
1864:
1841:
1834:
1819:
1810:
1808:
1802:that has an
1793:
1781:
1680:on a set as
1676:A group may
1675:
1660:
1645:
1620:homomorphism
1617:
1509:
1297:
1292:
1291:
1196:
1191:
1189:
1169:
1154:
940:
934:
912:, which has
907:
898:
887:left inverse
886:
841:
825:
812:
808:
801:
764:
657:
646:
580:
574:
572:
541:
525:
515:
511:
507:
504:
432:
401:proper class
399:(possibly a
390:
349:
345:
343:
248:
213:
204:
196:
180:
176:
162:
148:
144:
140:
134:
130:
126:
117:
106:left inverse
105:
96:
92:
88:
69:
63:
52:
42:
36:
5708:which is a
4244:determinant
3992:-semigroups
3941:-semigroups
3811:idempotents
3703:an element
3607:associative
2717:equals the
2493:associative
2486:composition
2462:Composition
2312:determinant
2298:. However,
2232:to itself.
2217:otherwise.
2125:is denoted
2069:associative
2065:commutative
1837:commutative
1636:isomorphism
1624:isomorphism
1371:commutative
1306:is denoted
1293:The inverse
1180:retractions
826:Similarly,
587:such that
435:associative
326:associative
240:isomorphism
169:associative
112:, and that
39:mathematics
5885:Categories
5811:References
4673:regression
4270:for more.
4089:such that
4027:involution
3970:-semigroup
3813:, that is
3719:such that
3707:is called
3183:invertible
3181:is called
2953:, we have
2921:. If, for
2810:surjective
2482:categories
1790:In monoids
1671:involution
1192:invertible
1165:surjective
1079:that maps
979:to either
647:for every
505:for every
58:reciprocal
5485:−
5458:−
5438:−
5423:−
5383:−
5373:−
5287:−
5229:−
5219:−
5185:−
5132:−
4915:−
4869:−
4776:×
4714:−
4603:−
4587:⏟
4578:−
4507:∣
4501:×
4447:−
4431:⏟
4412:−
4354:∣
4348:×
4325:full rank
4133:Semirings
3694:semigroup
3593:∗
3433:∗
3335:∗
3233:∗
2964:∗
2938:∈
2906:∈
2880:∗
2806:injective
2791:bijection
2755:→
2749::
2638:⊂
2581:∘
2552:→
2541::
2515:→
2509::
2478:morphisms
2335:rows and
2296:semirings
2196:zero ring
2138:−
1844:functions
1697:−
1642:In groups
1596:−
1583:−
1555:∗
1529:∗
1455:−
1452:∘
1419:−
1416:∗
1349:−
1314:−
1265:∗
1256:∗
1238:∗
1229:∗
1161:injective
1124:−
1042:↦
953:↦
937:functions
853:∗
784:∗
738:∗
677:∗
623:∗
601:∗
550:∗
487:∗
478:∗
460:∗
451:∗
411:∗
374:morphisms
328:and have
261:fractions
78:, and an
72:operation
70:Given an
5716:See also
4144:Examples
3953:for all
3861:, while
2730:identity
2695:codomain
2663:codomain
2645:′
2608:′
2548:′
2280:matrices
2271:Matrices
2050:addition
2036:In rings
2014:integers
2002:integers
1967:implies
1906:implies
1663:function
1654:with an
1632:morphism
1476:, since
1209:, then
1186:Inverses
1176:sections
1059:⌋
1046:⌊
581:identity
378:category
256:inversus
244:morphism
152:for all
82:denoted
47:opposite
5891:Algebra
4843:So, as
3743:inverse
3139:inverse
3113:, then
2985:, then
2892:and an
2869:with a
2818:modules
2282:over a
2265:subring
2228:from a
1626:. In
564:partial
181:inverse
5844:
5824:
5763:, and
3949:°)° =
3853:, and
3611:monoid
2780:object
2719:domain
2687:domain
2259:of an
2099:monoid
2044:is an
2024:, and
2020:of an
1796:monoid
920:, and
893:, and
877:where
358:domain
216:groups
191:, and
56:) and
5743:Notes
5480:right
4864:right
4847:<
4598:right
4217:field
4117:)* =
4109:)* =
4105:, (
4017:)° =
3793:then
3615:group
3141:, of
2863:magma
2824:or a
2464:is a
2343:is a
2321:is a
2288:rings
2284:field
2220:In a
2211:field
2109:. An
2012:from
2004:from
1826:units
1815:group
1667:arity
1650:is a
1648:group
897:is a
885:is a
395:is a
376:in a
251:Latin
224:units
220:rings
203:, an
116:is a
104:is a
86:, if
5842:ISBN
5822:ISBN
4513:<
4484:For
4442:left
4360:>
4331:For
4176:and
4164:and
4053:° =
3821:and
3809:are
3801:and
3757:and
3316:loop
3029:and
2841:Let
2530:and
2476:and
2402:in
2400:unit
2310:and
2308:rank
2294:and
2292:rngs
2115:unit
2067:and
2052:and
2042:ring
1935:and
1822:ring
1809:The
1514:and
1201:and
1075:the
769:and
662:and
651:and
542:Let
437:if
384:and
348:and
230:and
209:unit
201:ring
156:and
138:and
4749:No
4178:GLG
4170:LGL
4125:or
4113:, (
4099:yxy
4091:xyx
3957:in
3934:).
3779:zxz
3771:xzx
3763:yxy
3751:xyx
3749:if
3745:of
3721:xzx
3715:in
3670:or
3621:of
3605:is
3209:or
3185:in
3053:of
3009:of
2867:set
2808:or
2776:set
2721:of
2697:of
2689:of
2665:of
2498:If
2480:of
2472:of
2445:or
2421:If
2418:.
2390:An
2317:If
2205:If
2028:of
1678:act
1652:set
1510:If
1170:In
1167:.
1110:or
1083:to
1002:or
916:as
901:of
889:of
842:If
658:If
615:and
573:An
519:in
397:set
321:).
292:is
167:is
120:of
108:of
37:In
5887::
5866:,
5840:.
5836:.
5685:45
5680:36
5675:27
5668:36
5663:29
5658:22
5651:27
5646:22
5641:17
5453:13
5426:17
5410:18
5391:14
5386:32
5376:32
5368:77
5307:54
5237:14
5232:32
5222:32
5214:77
5201:54
5173:77
5168:32
5161:32
5156:14
5092:77
5087:32
5080:32
5075:14
4320:.
4192:A
4180:=
4172:=
4160:,
4129:.
4119:yx
4115:yx
4111:xy
4107:xy
4101:=
4097:,
4093:=
4051:aa
4015:ab
4002:=
3996:aa
3849:=
3847:fy
3845:=
3843:ye
3841:,
3837:=
3835:xf
3833:=
3831:ex
3825:=
3823:ff
3817:=
3815:ee
3807:yx
3805:=
3799:xy
3797:=
3777:=
3769:=
3761:=
3753:=
3727:;
3723:=
3696:.
3677:.
3576:3
3573:2
3570:4
3567:1
3564:5
3561:5
3556:1
3553:3
3550:2
3547:5
3544:4
3541:4
3536:2
3533:1
3530:5
3527:4
3524:3
3521:3
3516:4
3513:5
3510:1
3507:3
3504:2
3501:2
3496:5
3493:4
3490:3
3487:2
3484:1
3481:1
3476:5
3473:4
3470:3
3467:2
3464:1
3461:*
3452:.
3410:1
3407:2
3404:3
3401:3
3396:2
3393:1
3390:2
3387:2
3382:3
3379:2
3376:1
3373:1
3368:3
3365:2
3362:1
3359:*
3305:1
3302:1
3299:3
3296:3
3291:1
3288:1
3285:2
3282:2
3277:3
3274:2
3271:1
3268:1
3263:3
3260:2
3257:1
3254:*
3213:.
2801:.
2740:id
2725:.
2495:.
2447:−1
2387:.
2362:=
2355:AB
2314:.
2290:,
2235:A
2040:A
2032:.
2008:,
1994:).
1862:.
1820:A
1806:.
1794:A
1646:A
1638:.
1507:.
1182:.
905:.
762:)
570:.
534:.
514:,
510:,
368:,
253::
246:.
187:,
147:=
143:*
133:=
129:*
95:=
91:∗
62:1/
5850:.
5802:.
5771:.
5691:]
5635:[
5630:=
5625:]
5619:6
5614:5
5609:4
5602:3
5597:2
5592:1
5586:[
5579:]
5573:6
5568:3
5561:5
5556:2
5549:4
5544:1
5538:[
5533:=
5530:A
5525:T
5521:A
5488:1
5476:A
5472:=
5467:]
5461:4
5446:2
5441:2
5431:8
5417:[
5407:1
5402:=
5397:]
5362:[
5355:]
5349:6
5344:3
5337:5
5332:2
5325:4
5320:1
5314:[
5304:1
5299:=
5290:1
5282:)
5276:T
5272:A
5268:A
5264:(
5257:T
5253:A
5243:]
5208:[
5198:1
5193:=
5188:1
5179:]
5150:[
5144:=
5135:1
5127:)
5121:T
5117:A
5113:A
5109:(
5098:]
5069:[
5064:=
5059:]
5053:6
5048:3
5041:5
5036:2
5029:4
5024:1
5018:[
5011:]
5005:6
5000:5
4995:4
4988:3
4983:2
4978:1
4972:[
4967:=
4958:T
4954:A
4950:A
4923:.
4918:1
4910:)
4904:T
4900:A
4896:A
4892:(
4885:T
4881:A
4877:=
4872:1
4860:A
4849:n
4845:m
4826:]
4820:6
4815:5
4810:4
4803:3
4798:2
4793:1
4787:[
4782:=
4779:3
4773:2
4770::
4767:A
4735:.
4732:b
4727:T
4723:A
4717:1
4709:)
4705:A
4700:T
4696:A
4691:(
4686:=
4683:x
4655:b
4652:=
4649:x
4646:A
4621:m
4617:I
4613:=
4606:1
4594:A
4581:1
4573:)
4567:T
4563:A
4559:A
4555:(
4548:T
4544:A
4536:A
4516:n
4510:m
4504:n
4498:m
4495::
4492:A
4468:n
4464:I
4460:=
4457:A
4450:1
4438:A
4425:T
4421:A
4415:1
4407:)
4403:A
4398:T
4394:A
4389:(
4363:n
4357:m
4351:n
4345:m
4342::
4339:A
4308:R
4284:R
4254:M
4226:K
4203:M
4182:G
4174:L
4166:G
4162:L
4103:y
4095:x
4087:y
4083:x
4079:a
4063:a
4059:a
4057:°
4055:a
4047:I
4039:I
4033:*
4031:a
4023:a
4021:°
4019:b
4004:a
4000:a
3998:°
3990:I
3982:U
3978:a
3974:a
3968:U
3963:S
3959:S
3955:a
3951:a
3947:a
3939:U
3912:e
3909:H
3905:e
3900:1
3897:H
3888:e
3886:L
3881:e
3879:R
3875:e
3867:y
3863:f
3859:x
3855:e
3851:y
3839:x
3827:f
3819:e
3803:f
3795:e
3791:x
3787:y
3783:x
3775:y
3767:x
3759:y
3755:x
3747:x
3739:y
3729:z
3725:x
3717:S
3713:z
3705:x
3701:S
3675:1
3672:H
3658:)
3655:S
3652:(
3649:U
3629:S
3436:)
3430:,
3427:S
3424:(
3338:)
3332:,
3329:S
3326:(
3236:)
3230:,
3227:S
3224:(
3193:S
3169:S
3149:y
3121:x
3101:y
3081:x
3061:a
3037:b
3017:b
2993:a
2973:e
2970:=
2967:b
2961:a
2941:S
2935:b
2932:,
2929:a
2909:S
2903:e
2849:S
2774:(
2772:X
2758:X
2752:X
2744:X
2723:g
2705:f
2691:g
2673:f
2649:.
2642:Y
2635:Y
2615:Y
2612:=
2605:Y
2584:f
2578:g
2558:,
2555:Z
2545:Y
2538:g
2518:Y
2512:X
2506:f
2443:1
2423:R
2408:R
2404:R
2396:R
2385:0
2381:1
2364:p
2360:n
2350:q
2348:×
2346:p
2341:B
2337:n
2333:m
2328:n
2326:×
2324:m
2319:A
2245:0
2207:0
2200:0
2192:0
2176:.
2171:x
2168:1
2146:,
2141:1
2134:x
2123:x
2107:1
2091:x
2089:−
2081:x
2077:0
1982:z
1979:=
1976:y
1955:x
1952:z
1949:=
1946:x
1943:y
1923:,
1920:z
1917:=
1914:y
1894:z
1891:x
1888:=
1885:y
1882:x
1872:x
1768:.
1765:g
1745:,
1742:g
1722:g
1700:1
1693:g
1604:.
1599:1
1592:x
1586:1
1579:y
1558:y
1552:x
1532:y
1526:x
1516:y
1512:x
1489:f
1486:1
1458:1
1448:f
1427:.
1422:1
1412:x
1391:.
1386:x
1383:1
1357:,
1352:1
1345:x
1334:x
1320:.
1317:x
1304:x
1277:.
1274:r
1271:=
1268:r
1262:)
1259:x
1253:l
1250:(
1247:=
1244:)
1241:r
1235:x
1232:(
1226:l
1223:=
1220:l
1207:x
1203:r
1199:l
1150:n
1136:,
1131:2
1127:1
1121:n
1096:2
1093:n
1081:n
1063:,
1054:2
1051:n
1039:n
1019:1
1016:+
1013:n
1010:2
990:n
987:2
977:n
959:x
956:2
950:x
922:0
914:0
903:x
895:y
891:y
883:x
879:e
865:,
862:e
859:=
856:y
850:x
821:n
813:n
811:×
809:n
787:f
781:e
771:f
767:e
750:.
747:f
744:=
741:f
735:e
732:=
729:e
709:.
706:f
703:=
700:e
680:f
674:e
664:f
660:e
653:y
649:x
632:y
629:=
626:y
620:e
610:x
607:=
604:e
598:x
585:e
568:X
521:X
516:z
512:y
508:x
490:z
484:)
481:y
475:x
472:(
469:=
466:)
463:z
457:y
454:(
448:x
414:.
393:X
336:.
306:x
303:y
277:y
274:x
173:x
165:∗
158:y
154:x
149:y
145:y
141:e
135:x
131:e
127:x
122:x
114:y
110:y
102:x
97:e
93:y
89:x
84:e
76:∗
64:x
60:(
53:x
51:−
49:(
34:.
20:)
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