3252:
69:
3280:
2896:
1735:
20:
3247:{\displaystyle {\begin{aligned}L_{x}L_{x}^{-1}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad x(x\backslash y)&=y\\L_{x}^{-1}L_{x}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad x\backslash (xy)&=y\\R_{x}R_{x}^{-1}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad (y/x)x&=y\\R_{x}^{-1}R_{x}&=\mathrm {id} \qquad &{\text{corresponding to}}\qquad (yx)/x&=y\end{aligned}}}
4928:; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
3366:
under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in a
4389:
if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
5126:
3308:
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. See
2881:
4658:
Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup on
2738:
2302:
of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a
3598:
4133:
3538:
2901:
2774:
2655:
4568:
4192:
4327:
3876:
3834:
4023:
4379:
3649:
4945:
3739:
5552:
The first two equations are equivalent to the last two by direct application of the cancellation property of quasigroups. The last pair are shown to be equivalent by setting
3928:
3393:
4067:
2769:
4275:
4237:
3683:
1878:
3463:
3436:
5543:
Of these, the first three imply the last three, and vice versa, leading to either set of three identities being sufficient to equationally specify a quasigroup.
3968:
3948:
3779:
3759:
3355:. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.
640:
hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or
1145:
A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.
2650:
4711:
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation β (i.e.,
758:
are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called a
507:
98:
5407:
970:
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.
5222:
5212:
1815:. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e.
3544:
4767:
has two quasigroup operations, β and Β·, and one of them is isotopic to a conjugate of the other, the operations are said to be
551:
of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition.
4079:
6224:
6186:
6145:
6111:
5958:
5935:
5916:
1091:("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an
3482:
2119:(β) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity because
5200:
3310:
5643:
2886:
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on
6167:
4513:
4449:. Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
734:
satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.
500:
4393:
Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
4138:
120:
4755:
of β. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
91:
6274:
4280:
3305:
different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
731:
4747:
operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the
3839:
3784:
2532:
5431:
For clarity, cancellativity alone is insufficient: the requirement for existence of a solution must be retained.
40:
2277:
644:. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a
5615:
3973:
3329:, it is possible to imagine an infinite array in which every row and every column corresponds to some element
6257:
5121:{\displaystyle f(x_{1},\dots ,x_{n})=g(x_{1},\dots ,x_{i-1},\,h(x_{i},\dots ,x_{j}),\,x_{j+1},\dots ,x_{n}),}
4332:
493:
4651:
is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with the
3614:
3283:
A Latin square, the unbordered multiplication table for a quasigroup whose 10 elements are the digits 0β9.
6252:
759:
362:
4700:
3699:
766:
hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.
5694:
2876:{\displaystyle {\begin{aligned}L_{x}^{-1}(y)&=x\backslash y\\R_{x}^{-1}(y)&=y/x\end{aligned}}}
3888:
3374:
2298:
An associative quasigroup is either empty or is a group, since if there is at least one element, the
2291:
form a nonassociative loop under multiplication. The octonions are a special type of loop known as a
81:
6069:
4028:
6196:
Shcherbacov, V.A.; Pushkashu, D.I.; Shcherbacov, A.V. (2021). "Equational quasigroup definitions".
2167:
85:
77:
6241:
6247:
4939:
if its operation cannot be factored into the composition of two operations in the following way:
2618:
1746:
755:
453:
5653:
6284:
6064:
2599:
The Latin square property of quasigroups implies that, given any two of the three variables in
2152:
154:
102:
6279:
6125:
5412:
4242:
4204:
3396:
2541:
747:
657:
6212:
5906:
3658:
3371:. (This is somewhat misleading however, as the reals can be written in a sequence of length
1282:. This is equivalent to any one of the following single Moufang identities holding for all
6155:
5231:
3441:
3414:
3359:
751:
440:
432:
404:
399:
390:
347:
289:
6094:
5392:β an algebraic structure consisting of a set together with an associative binary operation
8:
6213:
5714:
5626:
5401:
4771:
to each other. There are also many other names for this relation of "isostrophe", e.g.,
3470:
2163:
2030:
1877:. Without idempotency, total symmetric quasigroups correspond to the geometric notion of
1111:
with the group identity (zero) turned into a "pointed idempotent". (That is, there is a
743:
548:
458:
448:
299:
199:
191:
182:
150:
146:
28:
6197:
6082:
6032:
6014:
5947:
5839:
5821:
4652:
3953:
3933:
3764:
3744:
3322:
1084:
264:
255:
213:
5766:
4931:
There exist multiary quasigroups that cannot be represented in any of these ways. An
699:. With regard to the Cayley table, the first equation (left division) means that the
6220:
6182:
6141:
6107:
5954:
5931:
5912:
5890:
5873:
5856:
4406:
763:
571:
537:
5843:
6133:
6090:
6074:
6036:
6024:
5987:
5831:
4585:. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
4458:
2517:
2273:
2102:
1018:
895:
algebra (i.e., equipped with three binary operations) that satisfy the identities:
530:
284:
162:
138:
56:
5740:
2733:{\displaystyle {\begin{aligned}L_{x}(y)&=xy\\R_{x}(y)&=yx\\\end{aligned}}}
2617:
The definition of a quasigroup can be treated as conditions on the left and right
309:
6151:
6121:
6086:
2307:
1077:
1073:
578:
525:
There are at least two structurally equivalent formal definitions of quasigroup:
376:
370:
357:
337:
328:
294:
231:
36:
24:
6049:
1836:
Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with)
6028:
5992:
5968:
5812:
Akivis, M.A.; Goldberg, Vladislav V. (2001). "Solution of
Belousov's problem".
2201:
2047:
2017:
1837:
1103:, taking its subtraction operation as quasigroup multiplication yields a pique
696:
418:
3279:
1852:
refers to an analogue for loops, namely, totally symmetric loops that satisfy
1148:
There are weaker associativity properties that have been given special names.
6268:
6163:
6045:
5902:
5894:
5860:
5383:
5181:
4696:
2763:, are bijective. The inverse mappings are left and right division, that is,
1092:
989:
is the same quasigroup in the sense of universal algebra. And vice versa: if
692:
581:, indicating that a quasigroup has to satisfy closure property), obeying the
304:
269:
226:
3688:
There are some stronger notions of inverses in loops that are often useful:
5877:
5682:
4692:
4616:
if there is an isotopy between them. In terms of Latin squares, an isotopy
4402:
3288:
3274:
2508:
2318:
2311:
2292:
2159:
1438:
1278:
645:
641:
545:
478:
409:
243:
6137:
981:
is a quasigroup according to the definition of the previous section, then
5185:
3363:
2505:
2209:
2175:
2116:
2109:
1441:, as their originators were studying quasigroups in Chicago at the time.
468:
463:
352:
342:
316:
158:
134:
1734:
157:" is always possible. Quasigroups differ from groups mainly in that the
6001:
Dudek, W.A.; Glazek, K. (2008). "Around the Hosszu-Gluskin
Theorem for
5835:
2205:
2171:
6078:
2520:
form a quasigroup with the operation of multiplication in the algebra.
6019:
5826:
5389:
5386:β a ring in which every non-zero element has a multiplicative inverse
4609:
2744:
2288:
727:
473:
279:
236:
204:
166:
6202:
4899:
An example of a multiary quasigroup is an iterated group operation,
3368:
1766:
1153:
274:
5889:(in Russian). Kishinev: Kishinev State University Printing House.
997:
is a quasigroup according to the sense of universal algebra, then
19:
5205:
The number of isomorphism classes of small quasigroups (sequence
4691:, but is not itself a group as it has no identity element. Every
2095:
711:
column while the second equation (right division) means that the
3476:
Every loop element has a unique left and right inverse given by
1437:
According to
Jonathan D. H. Smith, "loops" were named after the
6195:
5790:
5700:
5688:
5440:
There are six identities that these operations satisfy, namely
5395:
2280:
for its application. (The hyperbolic quaternions themselves do
656:
be unique can be replaced by the requirement that the magma be
208:
2755:
is a quasigroup precisely when all these operators, for every
5778:
4797:
5814:
Discussiones
Mathematicae β General Algebra and Applications
4853:
has a unique solution for any one variable if all the other
540:, defines a quasigroup as having three primitive operations.
5217:
5207:
2531:
In the remainder of the article we shall denote quasigroup
1453:) names the following important properties and subclasses:
5926:
Chein, O.; Pflugfelder, H.O.; Smith, J.D.H., eds. (1990).
3593:{\displaystyle x^{\rho }=x\backslash e\qquad xx^{\rho }=e}
3655:. In this case the inverse element is usually denoted by
6215:
5631:
5194:
4128:{\displaystyle (xy)^{\lambda }=y^{\lambda }x^{\lambda }}
691:. The operations '\' and '/' are called, respectively,
6126:"Example 4.1.3 (Zassenhaus's Commutative Moufang Loop)"
5658:
5368:
19,464,657,391,668,924,966,791,023,043,937,578,299,025
2564:. This follows from the uniqueness of left division of
1881:, also called Generalized Elliptic Cubic Curve (GECC).
1539:
Although this class may seem special, every quasigroup
3846:
3564:
3533:{\displaystyle x^{\lambda }=e/x\qquad x^{\lambda }x=e}
3043:
2965:
2814:
5925:
4948:
4516:
4335:
4283:
4245:
4207:
4141:
4082:
4031:
3976:
3956:
3936:
3891:
3842:
3787:
3767:
3747:
3702:
3661:
3617:
3547:
3485:
3444:
3417:
3377:
3287:
The multiplication table of a finite quasigroup is a
2899:
2772:
2653:
1465:
if any of the following equivalent identities hold:
663:
The unique solutions to these equations are written
6043:
5969:"Totally anti-symmetric quasigroups for all orders
5796:
5757:
5755:
5404:β has an additive and multiplicative loop structure
4783:
1276:A loop that is both a left and right Bol loop is a
1005:is a quasigroup according to the first definition.
5946:
5853:Foundations of the Theory of Quasigroups and Loops
5120:
4896:, or 2-ary, quasigroup is an ordinary quasigroup.
4640:, and a permutation on the underlying element set
4562:
4373:
4321:
4269:
4231:
4186:
4127:
4061:
4017:
3962:
3942:
3922:
3870:
3828:
3773:
3753:
3733:
3677:
3643:
3592:
3532:
3457:
3430:
3387:
3246:
2875:
2732:
2257:in the quasigroup. These quasigroups are known as
1844:, and sometimes the latter is even abbreviated as
750:is an equation in which all variables are tacitly
172:A quasigroup with an identity element is called a
6132:, Pure and Applied Mathematics, New York: Wiley,
5945:Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),
5670:
4563:{\displaystyle \alpha (x)\beta (y)=\gamma (xy)\,}
6266:
5752:
2310:. On the underlying set of the four-dimensional
90:but its sources remain unclear because it lacks
6120:
5887:Elements of Quasigroup Theory: a Special Course
5784:
5719:Recording of the Codes & Expansions Seminar
4592:is a homotopy for which each of the three maps
4187:{\displaystyle (xy)^{\rho }=y^{\rho }x^{\rho }}
2129:, but not a left identity because, in general,
2016:This property is required, for example, in the
6179:Elements of Quasigroup Theory and Applications
5944:
5928:Quasigroups and Loops: Theory and Applications
5811:
5772:
5169:; see Akivis and Goldberg (2001) for details.
4277:. This may be stated in terms of inverses via
2222:is the third element of the triple containing
1786:
1683:
6050:"Small Latin squares, quasigroups, and loops"
5701:Shcherbacov, Pushkashu & Shcherbacov 2021
5689:Shcherbacov, Pushkashu & Shcherbacov 2021
5408:Problems in loop theory and quasigroup theory
4322:{\displaystyle (xy)^{\lambda }x=y^{\lambda }}
3466:
2609:, the third variable is uniquely determined.
501:
5616:Nonempty associative quasigroup equals group
4706:
2516:More generally, the nonzero elements of any
2276:forms a nonassociative loop of order 8. See
6176:
6101:
6000:
5664:
4880:, quasigroup is just a constant element of
3871:{\displaystyle x\backslash y=x^{\lambda }y}
3829:{\displaystyle L_{x}^{-1}=L_{x^{\lambda }}}
2743:The definition says that both mappings are
2612:
529:One defines a quasigroup as a set with one
6161:
5872:(in Russian). Kishinev: Izdat. "Ε tiinca".
5649:
5357:2,750,892,211,809,150,446,995,735,533,513
3362:quasigroup, such as the group of non-zero
508:
494:
6201:
6068:
6018:
5991:
5825:
5076:
5034:
4559:
1112:
121:Learn how and when to remove this message
5884:
5867:
5850:
4758:
4018:{\displaystyle R_{x}^{-1}=R_{x^{\rho }}}
3407:The binary operation of a quasigroup is
3278:
1840:, so such a quasigroup is also called a
18:
4452:
4374:{\displaystyle x(yx)^{\rho }=y^{\rho }}
3467:left and right multiplication operators
3316:
2299:
1765:A quasigroup may exhibit semisymmetric
1068:, is unique, and that every element of
16:Magma obeying the Latin square property
6267:
6169:Equivalents of the Axiom of Choice, II
5855:(in Russian). Moscow: Izdat. "Nauka".
5742:Groups, Triality, and Hyperquasigroups
5398:β a semigroup with an identity element
5371:1,478,157,455,158,044,452,849,321,016
4857:variables are specified arbitrarily.
3644:{\displaystyle x^{\lambda }=x^{\rho }}
2887:
2533:multiplication simply by juxtaposition
2272:and with all other products as in the
1064:It follows that the identity element,
839:algebra that satisfy both identities:
784:algebra that satisfy both identities:
577:with a binary operation β (that is, a
6210:
5901:
5676:
5637:
5195:Number of small quasigroups and loops
3402:
2306:The following construction is due to
1884:
1450:
169:associative quasigroup equals group.
6044:McKay, Brendan D.; Meynert, Alison;
5966:
5775:, p. 497, definition 28.12
5761:
5627:an associative quasigroup is a group
1729:
737:
62:
6104:Quasigroups and Loops: Introduction
5201:Small Latin squares and quasigroups
4725:) we can form five new operations:
3380:
3311:small Latin squares and quasigroups
1985:, the following implication holds:
1919:, the following implication holds.
1543:induces a semisymmetric quasigroup
1252:) for each
13:
4869:-ary for some nonnegative integer
4778:
4683:is isotopic to the additive group
4632:is given by a permutation of rows
4422:between two quasigroups such that
3734:{\displaystyle x^{\lambda }(xy)=y}
3197:
3194:
3111:
3108:
3027:
3024:
2943:
2940:
1192: for each
1157:is a loop that satisfies either:
165:properties are optional. In fact,
14:
6296:
6235:
5949:Handbook of Combinatorial Designs
5797:McKay, Meynert & Myrvold 2007
5738:
5712:
2230:. These quasigroups also satisfy
1772:
5703:, p. 3, Thm. 1, 2
4784:Polyadic or multiary quasigroups
4074:antiautomorphic inverse property
3268:
3261:denotes the identity mapping on
1733:
67:
6124:; Smith, Jonathan D.H. (1999),
5732:
5706:
5546:
5434:
5346:15,224,734,061,438,247,321,497
5162:-ary quasigroups exist for all
4888:, quasigroup is a bijection of
3923:{\displaystyle (yx)x^{\rho }=y}
3570:
3510:
3388:{\displaystyle {\mathfrak {c}}}
3347:is in the row corresponding to
3209:
3201:
3123:
3115:
3039:
3031:
2955:
2947:
2178:quasigroup under the operation
1456:
5870:Algebraic Nets and Quasigroups
5620:
5609:
5425:
5112:
5070:
5038:
4993:
4984:
4952:
4556:
4547:
4538:
4532:
4526:
4520:
4349:
4339:
4294:
4284:
4258:
4249:
4217:
4208:
4152:
4142:
4093:
4083:
4062:{\displaystyle y/x=yx^{\rho }}
3901:
3892:
3722:
3713:
3219:
3210:
3138:
3124:
3055:
3046:
2971:
2959:
2848:
2842:
2801:
2795:
2710:
2704:
2674:
2668:
1080:(which need not be the same).
597:, there exist unique elements
520:
1:
5598:
4695:quasigroup is isotopic to an
4663:with multiplication given by
3351:and the column responding to
2524:
1899:weakly totally anti-symmetric
1684:conjugate division operations
1551:via the following operation:
1547:Ξ on the direct product cube
1444:
585:. This states that, for each
23:Algebraic structures between
5603:
4396:
1779:totally symmetric quasigroup
1682:where "//" and "\\" are the
44:
7:
6253:Encyclopedia of Mathematics
5953:(2nd ed.), CRC Press,
5785:Romanowska & Smith 1999
5377:
5360:20,890,436,195,945,769,617
4636:, a permutation of columns
3469:, are bijective, and hence
2284:form a loop or quasigroup.)
2023:
1789:coincide as one operation:
1725:
762:. Many standard results in
10:
6301:
6177:Shcherbacov, V.A. (2017).
6102:Pflugfelder, H.O. (1990).
6029:10.1016/j.disc.2007.09.005
5993:10.1016/j.disc.2006.05.033
5908:A Survey of Binary Systems
5804:
5773:Colbourn & Dinitz 2007
5715:"Codes, Errors, and Loops"
5198:
4456:
3272:
554:
48:
5967:Damm, H. Michael (2007).
4829:, such that the equation
4707:Conjugation (parastrophe)
1971:if, in addition, for all
648:.) The requirement that
5748:. Iowa State University.
5418:
5176:-ary quasigroup with an
3603:A loop is said to have (
3337:, and where the element
2619:multiplication operators
2613:Multiplication operators
2170:not equal to 2 forms an
1017:is a quasigroup with an
1008:
76:This article includes a
6275:Non-associative algebra
5885:Belousov, V.D. (1981).
5868:Belousov, V.D. (1971).
5851:Belousov, V.D. (1967).
4270:{\displaystyle x(yz)=e}
4232:{\displaystyle (xy)z=e}
3411:in the sense that both
1879:extended Steiner triple
1781:(sometimes abbreviated
1021:; that is, an element,
105:more precise citations.
6211:Smith, J.D.H. (2007).
6106:. Berlin: Heldermann.
5930:. Berlin: Heldermann.
5739:Smith, Jonathan D. H.
5713:Smith, Jonathan D. H.
5650:Rubin & Rubin 1985
5335:2,697,818,331,680,661
5236:Number of quasigroups
5215:) and loops (sequence
5158:. Finite irreducible
5122:
4612:. Two quasigroups are
4564:
4375:
4323:
4271:
4233:
4188:
4129:
4063:
4019:
3964:
3944:
3924:
3883:right inverse property
3872:
3830:
3775:
3755:
3735:
3679:
3678:{\displaystyle x^{-1}}
3645:
3594:
3534:
3459:
3432:
3389:
3284:
3248:
2877:
2734:
2278:hyperbolic quaternions
2155:(Γ·) form a quasigroup.
2147:(or the nonzero reals
2143:The nonzero rationals
1969:totally anti-symmetric
1777:A narrower class is a
752:universally quantified
732:empty binary operation
60:
6138:10.1002/9781118032589
5413:Mathematics of Sudoku
5123:
4759:Isostrophe (paratopy)
4565:
4401:A quasigroup or loop
4376:
4324:
4272:
4234:
4199:weak inverse property
4189:
4135:or, equivalently, if
4130:
4064:
4020:
3965:
3945:
3925:
3873:
3831:
3776:
3756:
3736:
3694:left inverse property
3680:
3646:
3595:
3535:
3460:
3458:{\displaystyle R_{x}}
3433:
3431:{\displaystyle L_{x}}
3397:well-ordering theorem
3390:
3282:
3249:
2878:
2735:
2542:cancellation property
2540:Quasigroups have the
2202:Steiner triple system
1083:A quasigroup with an
583:Latin square property
45:Latin square property
22:
5980:Discrete Mathematics
4946:
4701:BruckβToyoda theorem
4514:
4453:Homotopy and isotopy
4333:
4281:
4243:
4205:
4139:
4080:
4029:
3974:
3954:
3934:
3889:
3840:
3785:
3765:
3745:
3700:
3659:
3615:
3545:
3483:
3442:
3415:
3375:
3360:uncountably infinite
3317:Infinite quasigroups
2897:
2770:
2651:
2511:that is not a group.
405:Group with operators
348:Complemented lattice
183:Algebraic structures
6130:Post-modern algebra
6122:Romanowska, Anna B.
5640:, pp. 3, 26β27
5402:Planar ternary ring
4935:-ary quasigroup is
4473:quasigroup homotopy
3994:
3805:
3175:
3099:
3005:
2931:
2841:
2794:
2751:to itself. A magma
2317:over the 3-element
2259:Steiner quasigroups
2033:is a loop, because
754:, and in which all
744:algebraic structure
459:Composition algebra
219:Quasigroup and loop
153:in the sense that "
147:algebraic structure
5836:10.7151/dmgaa.1030
5349:9,365,022,303,540
5118:
4653:automorphism group
4560:
4471:be quasigroups. A
4371:
4319:
4267:
4229:
4184:
4125:
4059:
4015:
3977:
3960:
3940:
3920:
3868:
3826:
3788:
3771:
3751:
3731:
3675:
3641:
3590:
3530:
3455:
3428:
3403:Inverse properties
3385:
3323:countably infinite
3301:table filled with
3285:
3244:
3242:
3158:
3082:
2988:
2914:
2873:
2871:
2824:
2777:
2730:
2728:
1885:Total antisymmetry
1842:Steiner quasigroup
1745:. You can help by
1085:idempotent element
730:equipped with the
78:list of references
61:
6226:978-1-58488-537-5
6188:978-1-4987-2155-4
6147:978-0-471-12738-3
6113:978-3-88538-007-8
6079:10.1002/jcd.20105
5960:978-1-58488-506-1
5937:978-3-88538-008-5
5918:978-0-387-03497-3
5375:
5374:
5225:) is given here:
4796:is a set with an
3963:{\displaystyle y}
3943:{\displaystyle x}
3774:{\displaystyle y}
3754:{\displaystyle x}
3207:
3121:
3037:
2953:
2888:universal algebra
2536:
2270:ii = jj = kk = +1
1763:
1762:
1113:principal isotopy
764:universal algebra
738:Universal algebra
719:column is in the
538:universal algebra
518:
517:
131:
130:
123:
39:with the type of
6292:
6261:
6230:
6218:
6207:
6205:
6192:
6173:
6158:
6117:
6098:
6072:
6054:
6040:
6022:
5997:
5995:
5975:
5963:
5952:
5941:
5922:
5898:
5881:
5864:
5847:
5829:
5799:
5794:
5788:
5782:
5776:
5770:
5764:
5759:
5750:
5749:
5747:
5736:
5730:
5729:
5727:
5725:
5710:
5704:
5698:
5692:
5686:
5680:
5674:
5668:
5665:Pflugfelder 1990
5662:
5656:
5647:
5641:
5635:
5629:
5624:
5618:
5613:
5592:
5590:
5550:
5544:
5438:
5432:
5429:
5239:Number of loops
5228:
5227:
5220:
5210:
5180:-ary version of
5168:
5157:
5145:
5127:
5125:
5124:
5119:
5111:
5110:
5092:
5091:
5069:
5068:
5050:
5049:
5030:
5029:
5005:
5004:
4983:
4982:
4964:
4963:
4927:
4852:
4828:
4814:
4742:
4724:
4690:
4682:
4631:
4607:
4569:
4567:
4566:
4561:
4498:
4459:Isotopy of loops
4448:
4421:
4387:inverse property
4380:
4378:
4377:
4372:
4370:
4369:
4357:
4356:
4329:or equivalently
4328:
4326:
4325:
4320:
4318:
4317:
4302:
4301:
4276:
4274:
4273:
4268:
4238:
4236:
4235:
4230:
4193:
4191:
4190:
4185:
4183:
4182:
4173:
4172:
4160:
4159:
4134:
4132:
4131:
4126:
4124:
4123:
4114:
4113:
4101:
4100:
4068:
4066:
4065:
4060:
4058:
4057:
4039:
4024:
4022:
4021:
4016:
4014:
4013:
4012:
4011:
3993:
3985:
3970:. Equivalently,
3969:
3967:
3966:
3961:
3949:
3947:
3946:
3941:
3929:
3927:
3926:
3921:
3913:
3912:
3877:
3875:
3874:
3869:
3864:
3863:
3835:
3833:
3832:
3827:
3825:
3824:
3823:
3822:
3804:
3796:
3781:. Equivalently,
3780:
3778:
3777:
3772:
3760:
3758:
3757:
3752:
3740:
3738:
3737:
3732:
3712:
3711:
3684:
3682:
3681:
3676:
3674:
3673:
3650:
3648:
3647:
3642:
3640:
3639:
3627:
3626:
3599:
3597:
3596:
3591:
3583:
3582:
3557:
3556:
3539:
3537:
3536:
3531:
3520:
3519:
3506:
3495:
3494:
3464:
3462:
3461:
3456:
3454:
3453:
3437:
3435:
3434:
3429:
3427:
3426:
3394:
3392:
3391:
3386:
3384:
3383:
3346:
3300:
3260:
3253:
3251:
3250:
3245:
3243:
3226:
3208:
3206:corresponding to
3205:
3200:
3185:
3184:
3174:
3166:
3134:
3122:
3120:corresponding to
3119:
3114:
3098:
3090:
3081:
3080:
3038:
3036:corresponding to
3035:
3030:
3015:
3014:
3004:
2996:
2954:
2952:corresponding to
2951:
2946:
2930:
2922:
2913:
2912:
2882:
2880:
2879:
2874:
2872:
2865:
2840:
2832:
2793:
2785:
2739:
2737:
2736:
2731:
2729:
2703:
2702:
2667:
2666:
2643:
2608:
2595:
2585:
2576:. Similarly, if
2563:
2553:
2530:
2518:division algebra
2503:
2333:
2274:quaternion group
2271:
2267:
2266:{Β±1, Β±i, Β±j, Β±k}
2248:
2221:
2196:
2139:
2128:
2090:
2076:
2062:
2046:
1984:
1966:
1918:
1896:
1876:
1862:
1832:
1814:
1758:
1755:
1737:
1730:
1721:
1703:
1461:A quasigroup is
1151:For instance, a
1141:
1110:
1102:
1019:identity element
1004:
996:
988:
980:
894:
890:
838:
834:
783:
779:
771:right-quasigroup
690:
676:
569:
536:The other, from
531:binary operation
510:
503:
496:
285:Commutative ring
214:Rack and quandle
179:
178:
163:identity element
139:abstract algebra
137:, especially in
126:
119:
115:
112:
106:
101:this article by
92:inline citations
71:
70:
63:
57:identity element
6300:
6299:
6295:
6294:
6293:
6291:
6290:
6289:
6265:
6264:
6246:
6238:
6233:
6227:
6189:
6148:
6114:
6070:10.1.1.151.3043
6052:
6013:(21): 4861β76.
5970:
5961:
5938:
5919:
5807:
5802:
5795:
5791:
5783:
5779:
5771:
5767:
5760:
5753:
5745:
5737:
5733:
5723:
5721:
5711:
5707:
5699:
5695:
5687:
5683:
5675:
5671:
5663:
5659:
5648:
5644:
5636:
5632:
5625:
5621:
5614:
5610:
5606:
5601:
5596:
5595:
5553:
5551:
5547:
5439:
5435:
5430:
5426:
5421:
5380:
5324:12,198,455,835
5216:
5206:
5203:
5197:
5163:
5147:
5132:
5106:
5102:
5081:
5077:
5064:
5060:
5045:
5041:
5019:
5015:
5000:
4996:
4978:
4974:
4959:
4955:
4947:
4944:
4943:
4926:
4917:
4910:
4900:
4884:. A 1-ary, or
4846:
4840:
4830:
4816:
4804:
4786:
4781:
4779:Generalizations
4761:
4726:
4712:
4709:
4684:
4664:
4655:as a subgroup.
4617:
4593:
4515:
4512:
4511:
4484:
4461:
4455:
4423:
4409:
4399:
4385:A loop has the
4365:
4361:
4352:
4348:
4334:
4331:
4330:
4313:
4309:
4297:
4293:
4282:
4279:
4278:
4244:
4241:
4240:
4239:if and only if
4206:
4203:
4202:
4197:A loop has the
4178:
4174:
4168:
4164:
4155:
4151:
4140:
4137:
4136:
4119:
4115:
4109:
4105:
4096:
4092:
4081:
4078:
4077:
4072:A loop has the
4053:
4049:
4035:
4030:
4027:
4026:
4007:
4003:
4002:
3998:
3986:
3981:
3975:
3972:
3971:
3955:
3952:
3951:
3935:
3932:
3931:
3908:
3904:
3890:
3887:
3886:
3881:A loop has the
3859:
3855:
3841:
3838:
3837:
3818:
3814:
3813:
3809:
3797:
3792:
3786:
3783:
3782:
3766:
3763:
3762:
3746:
3743:
3742:
3707:
3703:
3701:
3698:
3697:
3692:A loop has the
3666:
3662:
3660:
3657:
3656:
3635:
3631:
3622:
3618:
3616:
3613:
3612:
3578:
3574:
3552:
3548:
3546:
3543:
3542:
3515:
3511:
3502:
3490:
3486:
3484:
3481:
3480:
3449:
3445:
3443:
3440:
3439:
3422:
3418:
3416:
3413:
3412:
3405:
3395:, assuming the
3379:
3378:
3376:
3373:
3372:
3338:
3319:
3292:
3277:
3271:
3258:
3241:
3240:
3230:
3222:
3204:
3202:
3193:
3186:
3180:
3176:
3167:
3162:
3155:
3154:
3144:
3130:
3118:
3116:
3107:
3100:
3091:
3086:
3076:
3072:
3069:
3068:
3058:
3034:
3032:
3023:
3016:
3010:
3006:
2997:
2992:
2985:
2984:
2974:
2950:
2948:
2939:
2932:
2923:
2918:
2908:
2904:
2900:
2898:
2895:
2894:
2870:
2869:
2861:
2851:
2833:
2828:
2821:
2820:
2804:
2786:
2781:
2773:
2771:
2768:
2767:
2727:
2726:
2713:
2698:
2694:
2691:
2690:
2677:
2662:
2658:
2654:
2652:
2649:
2648:
2634:
2627:
2621:
2615:
2600:
2587:
2577:
2555:
2545:
2527:
2497:
2488:
2482:
2475:
2469:
2462:
2455:
2449:) + (0, 0, 0, (
2448:
2441:
2434:
2427:
2420:
2413:
2406:
2399:
2392:
2385:
2378:
2371:
2364:
2357:
2350:
2343:
2321:
2308:Hans Zassenhaus
2269:
2265:
2231:
2213:
2179:
2130:
2120:
2078:
2077:if and only if
2064:
2050:
2034:
2026:
1972:
1960:
1902:
1890:
1887:
1864:
1853:
1838:Steiner triples
1816:
1790:
1785:) in which all
1775:
1759:
1753:
1750:
1743:needs expansion
1728:
1705:
1687:
1677:
1670:
1666:
1659:
1652:
1645:
1638:
1631:
1624:
1617:
1610:
1603:
1596:
1589:
1582:
1575:
1568:
1561:
1459:
1447:
1115:
1104:
1096:
1011:
998:
990:
982:
974:
892:
884:
836:
828:
826:left-quasigroup
781:
773:
740:
678:
664:
609:such that both
570:is a non-empty
563:
557:
523:
514:
485:
484:
483:
454:Non-associative
436:
425:
424:
414:
394:
383:
382:
371:Map of lattices
367:
363:Boolean algebra
358:Heyting algebra
332:
321:
320:
314:
295:Integral domain
259:
248:
247:
241:
195:
127:
116:
110:
107:
96:
82:related reading
72:
68:
17:
12:
11:
5:
6298:
6288:
6287:
6282:
6277:
6263:
6262:
6244:
6237:
6236:External links
6234:
6232:
6231:
6225:
6208:
6193:
6187:
6174:
6159:
6146:
6118:
6112:
6099:
6046:Myrvold, Wendy
6041:
6005:-ary groups".
5998:
5986:(6): 715β729.
5964:
5959:
5942:
5936:
5923:
5917:
5899:
5882:
5865:
5848:
5808:
5806:
5803:
5801:
5800:
5789:
5777:
5765:
5751:
5731:
5705:
5693:
5681:
5669:
5657:
5642:
5630:
5619:
5607:
5605:
5602:
5600:
5597:
5594:
5593:
5545:
5542:
5541:
5491:
5433:
5423:
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5410:
5405:
5399:
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5387:
5379:
5376:
5373:
5372:
5369:
5366:
5362:
5361:
5358:
5355:
5351:
5350:
5347:
5344:
5340:
5339:
5336:
5333:
5329:
5328:
5325:
5322:
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5311:
5307:
5306:
5303:
5300:
5296:
5295:
5292:
5289:
5285:
5284:
5281:
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5274:
5273:
5270:
5267:
5263:
5262:
5259:
5256:
5252:
5251:
5248:
5245:
5241:
5240:
5237:
5234:
5199:Main article:
5196:
5193:
5129:
5128:
5117:
5114:
5109:
5105:
5101:
5098:
5095:
5090:
5087:
5084:
5080:
5075:
5072:
5067:
5063:
5059:
5056:
5053:
5048:
5044:
5040:
5037:
5033:
5028:
5025:
5022:
5018:
5014:
5011:
5008:
5003:
4999:
4995:
4992:
4989:
4986:
4981:
4977:
4973:
4970:
4967:
4962:
4958:
4954:
4951:
4922:
4915:
4908:
4892:to itself. A
4844:
4838:
4801:-ary operation
4794:ary quasigroup
4785:
4782:
4780:
4777:
4760:
4757:
4708:
4705:
4571:
4570:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4457:Main article:
4454:
4451:
4398:
4395:
4383:
4382:
4368:
4364:
4360:
4355:
4351:
4347:
4344:
4341:
4338:
4316:
4312:
4308:
4305:
4300:
4296:
4292:
4289:
4286:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4195:
4181:
4177:
4171:
4167:
4163:
4158:
4154:
4150:
4147:
4144:
4122:
4118:
4112:
4108:
4104:
4099:
4095:
4091:
4088:
4085:
4070:
4056:
4052:
4048:
4045:
4042:
4038:
4034:
4010:
4006:
4001:
3997:
3992:
3989:
3984:
3980:
3959:
3939:
3919:
3916:
3911:
3907:
3903:
3900:
3897:
3894:
3879:
3867:
3862:
3858:
3854:
3851:
3848:
3845:
3821:
3817:
3812:
3808:
3803:
3800:
3795:
3791:
3770:
3750:
3730:
3727:
3724:
3721:
3718:
3715:
3710:
3706:
3672:
3669:
3665:
3638:
3634:
3630:
3625:
3621:
3601:
3600:
3589:
3586:
3581:
3577:
3573:
3569:
3566:
3563:
3560:
3555:
3551:
3540:
3529:
3526:
3523:
3518:
3514:
3509:
3505:
3501:
3498:
3493:
3489:
3452:
3448:
3425:
3421:
3404:
3401:
3382:
3318:
3315:
3273:Main article:
3270:
3267:
3255:
3254:
3239:
3236:
3233:
3231:
3229:
3225:
3221:
3218:
3215:
3212:
3203:
3199:
3196:
3192:
3189:
3187:
3183:
3179:
3173:
3170:
3165:
3161:
3157:
3156:
3153:
3150:
3147:
3145:
3143:
3140:
3137:
3133:
3129:
3126:
3117:
3113:
3110:
3106:
3103:
3101:
3097:
3094:
3089:
3085:
3079:
3075:
3071:
3070:
3067:
3064:
3061:
3059:
3057:
3054:
3051:
3048:
3045:
3042:
3033:
3029:
3026:
3022:
3019:
3017:
3013:
3009:
3003:
3000:
2995:
2991:
2987:
2986:
2983:
2980:
2977:
2975:
2973:
2970:
2967:
2964:
2961:
2958:
2949:
2945:
2942:
2938:
2935:
2933:
2929:
2926:
2921:
2917:
2911:
2907:
2903:
2902:
2884:
2883:
2868:
2864:
2860:
2857:
2854:
2852:
2850:
2847:
2844:
2839:
2836:
2831:
2827:
2823:
2822:
2819:
2816:
2813:
2810:
2807:
2805:
2803:
2800:
2797:
2792:
2789:
2784:
2780:
2776:
2775:
2741:
2740:
2725:
2722:
2719:
2716:
2714:
2712:
2709:
2706:
2701:
2697:
2693:
2692:
2689:
2686:
2683:
2680:
2678:
2676:
2673:
2670:
2665:
2661:
2657:
2656:
2644:, defined by
2632:
2625:
2614:
2611:
2538:
2537:
2526:
2523:
2522:
2521:
2513:
2512:
2493:
2492:
2491:
2490:
2486:
2480:
2473:
2467:
2460:
2453:
2446:
2439:
2432:
2425:
2418:
2411:
2404:
2397:
2390:
2383:
2376:
2369:
2362:
2355:
2348:
2341:
2304:
2296:
2285:
2262:
2198:
2168:characteristic
2156:
2141:
2092:
2048:if and only if
2025:
2022:
2018:Damm algorithm
2014:
2013:
1957:
1956:
1886:
1883:
1774:
1773:Total symmetry
1771:
1761:
1760:
1740:
1738:
1727:
1724:
1680:
1679:
1675:
1668:
1664:
1657:
1650:
1643:
1636:
1629:
1622:
1615:
1608:
1601:
1594:
1587:
1580:
1573:
1566:
1559:
1537:
1536:
1518:
1501:
1484:
1458:
1455:
1446:
1443:
1435:
1434:
1399:
1364:
1330:
1274:
1273:
1270:right Bol loop
1214:
1213:
1078:right inverses
1062:
1061:
1010:
1007:
968:
967:
949:
931:
914:
878:
877:
859:
822:
821:
803:
739:
736:
707:row is in the
697:right division
638:
637:
624:
556:
553:
542:
541:
534:
522:
519:
516:
515:
513:
512:
505:
498:
490:
487:
486:
482:
481:
476:
471:
466:
461:
456:
451:
445:
444:
443:
437:
431:
430:
427:
426:
423:
422:
419:Linear algebra
413:
412:
407:
402:
396:
395:
389:
388:
385:
384:
381:
380:
377:Lattice theory
373:
366:
365:
360:
355:
350:
345:
340:
334:
333:
327:
326:
323:
322:
313:
312:
307:
302:
297:
292:
287:
282:
277:
272:
267:
261:
260:
254:
253:
250:
249:
240:
239:
234:
229:
223:
222:
221:
216:
211:
202:
196:
190:
189:
186:
185:
129:
128:
86:external links
75:
73:
66:
15:
9:
6:
4:
3:
2:
6297:
6286:
6285:Latin squares
6283:
6281:
6278:
6276:
6273:
6272:
6270:
6259:
6255:
6254:
6249:
6248:"Quasi-group"
6245:
6243:
6240:
6239:
6228:
6222:
6219:. CRC Press.
6217:
6216:
6209:
6204:
6199:
6194:
6190:
6184:
6181:. CRC Press.
6180:
6175:
6171:
6170:
6165:
6160:
6157:
6153:
6149:
6143:
6139:
6135:
6131:
6127:
6123:
6119:
6115:
6109:
6105:
6100:
6096:
6092:
6088:
6084:
6080:
6076:
6071:
6066:
6063:(2): 98β119.
6062:
6058:
6051:
6047:
6042:
6038:
6034:
6030:
6026:
6021:
6016:
6012:
6008:
6007:Discrete Math
6004:
5999:
5994:
5989:
5985:
5981:
5977:
5973:
5965:
5962:
5956:
5951:
5950:
5943:
5939:
5933:
5929:
5924:
5920:
5914:
5910:
5909:
5904:
5900:
5896:
5892:
5888:
5883:
5879:
5875:
5871:
5866:
5862:
5858:
5854:
5849:
5845:
5841:
5837:
5833:
5828:
5823:
5820:(1): 93β103.
5819:
5815:
5810:
5809:
5798:
5793:
5786:
5781:
5774:
5769:
5763:
5758:
5756:
5744:
5743:
5735:
5720:
5716:
5709:
5702:
5697:
5690:
5685:
5678:
5673:
5666:
5661:
5655:
5651:
5646:
5639:
5634:
5628:
5623:
5617:
5612:
5608:
5588:
5584:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5549:
5539:
5535:
5531:
5527:
5523:
5519:
5515:
5511:
5507:
5503:
5499:
5495:
5492:
5489:
5485:
5481:
5477:
5473:
5469:
5465:
5461:
5457:
5453:
5449:
5445:
5442:
5441:
5437:
5428:
5424:
5414:
5411:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5384:Division ring
5382:
5381:
5370:
5367:
5364:
5363:
5359:
5356:
5353:
5352:
5348:
5345:
5342:
5341:
5337:
5334:
5331:
5330:
5326:
5323:
5320:
5319:
5315:
5312:
5309:
5308:
5304:
5301:
5298:
5297:
5293:
5290:
5287:
5286:
5282:
5279:
5276:
5275:
5271:
5268:
5265:
5264:
5260:
5257:
5254:
5253:
5249:
5246:
5243:
5242:
5238:
5235:
5233:
5230:
5229:
5226:
5224:
5219:
5214:
5209:
5202:
5192:
5190:
5188:
5184:is called an
5183:
5182:associativity
5179:
5175:
5170:
5166:
5161:
5155:
5151:
5144:
5140:
5136:
5115:
5107:
5103:
5099:
5096:
5093:
5088:
5085:
5082:
5078:
5073:
5065:
5061:
5057:
5054:
5051:
5046:
5042:
5035:
5031:
5026:
5023:
5020:
5016:
5012:
5009:
5006:
5001:
4997:
4990:
4987:
4979:
4975:
4971:
4968:
4965:
4960:
4956:
4949:
4942:
4941:
4940:
4938:
4934:
4929:
4925:
4921:
4914:
4907:
4903:
4897:
4895:
4891:
4887:
4883:
4879:
4874:
4872:
4868:
4864:
4860:
4856:
4851:
4847:
4837:
4833:
4827:
4823:
4819:
4812:
4808:
4802:
4800:
4795:
4791:
4776:
4774:
4770:
4766:
4756:
4754:
4750:
4746:
4741:
4737:
4733:
4729:
4723:
4719:
4715:
4704:
4702:
4698:
4697:abelian group
4694:
4688:
4680:
4676:
4672:
4668:
4662:
4656:
4654:
4650:
4645:
4643:
4639:
4635:
4629:
4625:
4621:
4615:
4611:
4605:
4601:
4597:
4591:
4586:
4584:
4580:
4576:
4553:
4550:
4544:
4541:
4535:
4529:
4523:
4517:
4510:
4509:
4508:
4506:
4502:
4499:of maps from
4496:
4492:
4488:
4482:
4478:
4474:
4470:
4466:
4460:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4420:
4416:
4412:
4408:
4404:
4394:
4391:
4388:
4366:
4362:
4358:
4353:
4345:
4342:
4336:
4314:
4310:
4306:
4303:
4298:
4290:
4287:
4264:
4261:
4255:
4252:
4246:
4226:
4223:
4220:
4214:
4211:
4200:
4196:
4179:
4175:
4169:
4165:
4161:
4156:
4148:
4145:
4120:
4116:
4110:
4106:
4102:
4097:
4089:
4086:
4075:
4071:
4054:
4050:
4046:
4043:
4040:
4036:
4032:
4008:
4004:
3999:
3995:
3990:
3987:
3982:
3978:
3957:
3937:
3917:
3914:
3909:
3905:
3898:
3895:
3884:
3880:
3865:
3860:
3856:
3852:
3849:
3843:
3819:
3815:
3810:
3806:
3801:
3798:
3793:
3789:
3768:
3748:
3728:
3725:
3719:
3716:
3708:
3704:
3695:
3691:
3690:
3689:
3686:
3670:
3667:
3663:
3654:
3636:
3632:
3628:
3623:
3619:
3610:
3606:
3587:
3584:
3579:
3575:
3571:
3567:
3561:
3558:
3553:
3549:
3541:
3527:
3524:
3521:
3516:
3512:
3507:
3503:
3499:
3496:
3491:
3487:
3479:
3478:
3477:
3474:
3472:
3468:
3450:
3446:
3423:
3419:
3410:
3400:
3398:
3370:
3365:
3361:
3356:
3354:
3350:
3345:
3341:
3336:
3332:
3328:
3324:
3314:
3312:
3306:
3304:
3299:
3295:
3290:
3281:
3276:
3269:Latin squares
3266:
3264:
3237:
3234:
3232:
3227:
3223:
3216:
3213:
3190:
3188:
3181:
3177:
3171:
3168:
3163:
3159:
3151:
3148:
3146:
3141:
3135:
3131:
3127:
3104:
3102:
3095:
3092:
3087:
3083:
3077:
3073:
3065:
3062:
3060:
3052:
3049:
3040:
3020:
3018:
3011:
3007:
3001:
2998:
2993:
2989:
2981:
2978:
2976:
2968:
2962:
2956:
2936:
2934:
2927:
2924:
2919:
2915:
2909:
2905:
2893:
2892:
2891:
2889:
2866:
2862:
2858:
2855:
2853:
2845:
2837:
2834:
2829:
2825:
2817:
2811:
2808:
2806:
2798:
2790:
2787:
2782:
2778:
2766:
2765:
2764:
2762:
2758:
2754:
2750:
2746:
2723:
2720:
2717:
2715:
2707:
2699:
2695:
2687:
2684:
2681:
2679:
2671:
2663:
2659:
2647:
2646:
2645:
2642:
2638:
2631:
2624:
2620:
2610:
2607:
2603:
2597:
2594:
2590:
2584:
2580:
2575:
2571:
2567:
2562:
2558:
2552:
2548:
2543:
2534:
2529:
2528:
2519:
2515:
2514:
2510:
2507:
2501:
2495:
2494:
2485:
2479:
2472:
2466:
2459:
2452:
2445:
2438:
2431:
2424:
2417:
2410:
2403:
2396:
2389:
2382:
2375:
2368:
2361:
2354:
2347:
2340:
2336:
2335:
2332:
2328:
2324:
2320:
2316:
2313:
2309:
2305:
2301:
2300:invertibility
2297:
2294:
2290:
2286:
2283:
2279:
2275:
2263:
2260:
2256:
2252:
2247:
2243:
2239:
2235:
2229:
2225:
2220:
2216:
2211:
2207:
2203:
2199:
2194:
2190:
2186:
2182:
2177:
2173:
2169:
2165:
2161:
2157:
2154:
2150:
2146:
2142:
2138:
2134:
2127:
2123:
2118:
2114:
2111:
2107:
2104:
2100:
2097:
2093:
2089:
2085:
2081:
2075:
2071:
2067:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2032:
2028:
2027:
2021:
2019:
2011:
2007:
2004:implies that
2003:
1999:
1995:
1991:
1988:
1987:
1986:
1983:
1979:
1975:
1970:
1964:
1959:A quasigroup
1954:
1950:
1947:implies that
1946:
1942:
1938:
1934:
1930:
1926:
1922:
1921:
1920:
1917:
1913:
1909:
1905:
1900:
1894:
1889:A quasigroup
1882:
1880:
1875:
1871:
1867:
1860:
1856:
1851:
1847:
1843:
1839:
1834:
1831:
1827:
1823:
1819:
1813:
1809:
1805:
1801:
1797:
1793:
1788:
1784:
1783:TS-quasigroup
1780:
1770:
1768:
1757:
1754:February 2015
1748:
1744:
1741:This section
1739:
1736:
1732:
1731:
1723:
1720:
1716:
1712:
1708:
1702:
1698:
1694:
1690:
1685:
1674:
1663:
1656:
1649:
1642:
1635:
1628:
1621:
1614:
1607:
1600:
1593:
1586:
1579:
1572:
1565:
1558:
1554:
1553:
1552:
1550:
1546:
1542:
1534:
1530:
1526:
1522:
1519:
1517:
1513:
1509:
1505:
1502:
1500:
1496:
1492:
1488:
1485:
1483:
1479:
1475:
1471:
1468:
1467:
1466:
1464:
1463:semisymmetric
1454:
1452:
1442:
1440:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1331:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1297:
1296:
1295:
1293:
1289:
1285:
1281:
1280:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1218:
1217:
1211:
1210:left Bol loop
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1160:
1159:
1158:
1156:
1155:
1149:
1146:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1114:
1108:
1100:
1094:
1093:abelian group
1090:
1086:
1081:
1079:
1075:
1071:
1067:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1028:
1027:
1026:
1024:
1020:
1016:
1006:
1002:
994:
986:
978:
971:
965:
961:
957:
953:
950:
947:
943:
939:
935:
932:
930:
926:
922:
918:
915:
913:
909:
905:
901:
898:
897:
896:
888:
883:
875:
871:
867:
863:
860:
857:
853:
849:
845:
842:
841:
840:
832:
827:
819:
815:
811:
807:
804:
802:
798:
794:
790:
787:
786:
785:
777:
772:
767:
765:
761:
757:
753:
749:
745:
735:
733:
729:
724:
722:
718:
715:entry in the
714:
710:
706:
703:entry in the
702:
698:
694:
693:left division
689:
685:
681:
675:
671:
667:
661:
659:
655:
651:
647:
643:
636:
632:
628:
625:
623:
619:
615:
612:
611:
610:
608:
604:
600:
596:
592:
588:
584:
580:
576:
573:
567:
562:
552:
550:
547:
539:
535:
532:
528:
527:
526:
511:
506:
504:
499:
497:
492:
491:
489:
488:
480:
477:
475:
472:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
446:
442:
439:
438:
434:
429:
428:
421:
420:
416:
415:
411:
408:
406:
403:
401:
398:
397:
392:
387:
386:
379:
378:
374:
372:
369:
368:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
335:
330:
325:
324:
319:
318:
311:
308:
306:
305:Division ring
303:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
262:
257:
252:
251:
246:
245:
238:
235:
233:
230:
228:
227:Abelian group
225:
224:
220:
217:
215:
212:
210:
206:
203:
201:
198:
197:
193:
188:
187:
184:
181:
180:
177:
175:
170:
168:
164:
160:
156:
152:
149:resembling a
148:
144:
140:
136:
125:
122:
114:
111:February 2024
104:
100:
94:
93:
87:
83:
79:
74:
65:
64:
58:
54:
50:
46:
43:given by the
42:
38:
34:
30:
26:
21:
6280:Group theory
6251:
6214:
6178:
6168:
6129:
6103:
6060:
6057:J. Comb. Des
6056:
6020:math/0510185
6010:
6006:
6002:
5983:
5979:
5971:
5948:
5927:
5911:. Springer.
5907:
5886:
5869:
5852:
5827:math/0010175
5817:
5813:
5792:
5787:, p. 93
5780:
5768:
5741:
5734:
5722:. Retrieved
5718:
5708:
5696:
5684:
5672:
5660:
5645:
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5467:
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5338:106,228,849
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4930:
4923:
4919:
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4889:
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4877:
4876:A 0-ary, or
4875:
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4753:parastrophes
4752:
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4483:is a triple
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4403:homomorphism
4400:
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4073:
3882:
3693:
3687:
3652:
3608:
3604:
3602:
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3406:
3364:real numbers
3357:
3352:
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3326:
3320:
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3302:
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3293:
3289:Latin square
3286:
3275:Latin square
3262:
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2588:
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2565:
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2509:Moufang loop
2499:
2483:
2477:
2470:
2464:
2457:
2450:
2443:
2436:
2429:
2422:
2415:
2408:
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2373:
2366:
2359:
2352:
2345:
2338:
2330:
2326:
2322:
2319:Galois field
2314:
2312:vector space
2293:Moufang loop
2287:The nonzero
2281:
2258:
2254:
2250:
2245:
2241:
2237:
2233:
2227:
2223:
2218:
2214:
2212:quasigroup:
2192:
2188:
2184:
2180:
2160:vector space
2148:
2144:
2136:
2132:
2125:
2121:
2112:
2105:
2098:
2087:
2083:
2079:
2073:
2069:
2065:
2059:
2055:
2051:
2043:
2039:
2035:
2015:
2009:
2005:
2001:
1997:
1993:
1989:
1981:
1977:
1973:
1968:
1962:
1958:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1915:
1911:
1907:
1903:
1898:
1892:
1888:
1873:
1869:
1865:
1858:
1854:
1849:
1848:. The term
1845:
1841:
1835:
1829:
1825:
1821:
1817:
1811:
1807:
1803:
1799:
1795:
1791:
1782:
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1776:
1764:
1751:
1747:adding to it
1742:
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1672:
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1626:
1619:
1612:
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1524:
1520:
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1494:
1490:
1486:
1481:
1477:
1473:
1469:
1462:
1460:
1457:Semisymmetry
1448:
1439:Chicago Loop
1436:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1395:
1391:
1387:
1383:
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1375:
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1326:
1322:
1318:
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1310:
1306:
1302:
1298:
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1287:
1283:
1279:Moufang loop
1277:
1275:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
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1088:
1087:is called a
1082:
1069:
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1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:, such that
1022:
1014:
1012:
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972:
969:
963:
959:
955:
951:
945:
941:
937:
933:
928:
924:
920:
916:
911:
907:
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899:
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879:
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869:
865:
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847:
843:
830:
825:
823:
817:
813:
809:
805:
800:
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792:
788:
775:
770:
768:
741:
725:
720:
716:
712:
708:
704:
700:
687:
683:
679:
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669:
665:
662:
658:cancellative
653:
649:
646:Latin square
642:Cayley table
639:
634:
630:
626:
621:
617:
613:
606:
602:
598:
594:
590:
586:
582:
574:
565:
560:
558:
543:
524:
479:Hopf algebra
417:
410:Vector space
375:
315:
244:Group theory
242:
218:
207: /
173:
171:
142:
132:
117:
108:
97:Please help
89:
52:
41:divisibility
32:
6242:quasigroups
6203:1003.3175v1
6172:. Elsevier.
6164:Rubin, J.E.
6162:Rubin, H.;
5903:Bruck, R.H.
5691:, p. 1
5679:, p. 1
5667:, p. 2
4937:irreducible
4769:isostrophic
4763:If the set
3325:quasigroup
2506:commutative
2210:commutative
2204:defines an
2176:commutative
2117:subtraction
1901:if for all
1863:instead of
1072:has unique
742:Given some
546:homomorphic
521:Definitions
464:Lie algebra
449:Associative
353:Total order
343:Semilattice
317:Ring theory
159:associative
135:mathematics
103:introducing
6269:Categories
6095:1112.05018
5677:Bruck 1971
5652:, p.
5638:Smith 2007
5599:References
5313:1,130,531
5189:-ary group
4749:conjugates
4507:such that
3471:invertible
3409:invertible
2745:bijections
2525:Properties
2206:idempotent
2172:idempotent
1967:is called
1897:is called
1787:conjugates
1451:Smith 2007
1445:Symmetries
995:, β, \, /)
987:, β, \, /)
891:is a type
889:, β, \, /)
882:quasigroup
835:is a type
780:is a type
756:operations
561:quasigroup
143:quasigroup
53:quasigroup
33:quasigroup
6258:EMS Press
6065:CiteSeerX
5905:(1971) .
5895:318458899
5861:472241611
5762:Damm 2007
5604:Citations
5390:Semigroup
5097:…
5055:…
5024:−
5010:…
4969:…
4734: :=
4610:bijection
4545:γ
4530:β
4518:α
4397:Morphisms
4367:ρ
4354:ρ
4315:λ
4299:λ
4180:ρ
4170:ρ
4157:ρ
4121:λ
4111:λ
4098:λ
4055:ρ
4009:ρ
3988:−
3910:ρ
3861:λ
3847:∖
3820:λ
3799:−
3709:λ
3668:−
3637:ρ
3624:λ
3605:two-sided
3580:ρ
3565:∖
3554:ρ
3517:λ
3492:λ
3169:−
3093:−
3044:∖
2999:−
2966:∖
2925:−
2835:−
2815:∖
2788:−
2289:octonions
2103:rationals
1686:given by
973:Hence if
893:(2, 2, 2)
728:empty set
474:Bialgebra
280:Near-ring
237:Lie group
205:Semigroup
6166:(1985).
6048:(2007).
5844:18421746
5378:See also
5152:) β (1,
4918:Β· Β·Β·Β· Β·
4863:multiary
4859:Polyadic
4820: :
4773:paratopy
4745:opposite
4649:autotopy
4614:isotopic
4573:for all
4413: :
3930:for all
3741:for all
3651:for all
3609:inverses
3369:sequence
2635: :
2264:The set
2249:for all
2153:division
2101:(or the
2096:integers
2024:Examples
1767:triality
1726:Triality
1216:or else
1154:Bol loop
1052:for all
748:identity
310:Lie ring
275:Semiring
167:nonempty
155:division
55:with an
6260:, 2001
6156:1673047
6037:9545943
5878:8292276
5805:Sources
5724:2 April
5327:23,746
5221:in the
5218:A057771
5211:in the
5208:A057991
4878:nullary
4699:by the
4590:isotopy
3358:For an
2890:) are
2586:, then
2554:, then
2334:define
2162:over a
2151:) with
2115:) with
2108:or the
1347:)) = ((
1313:)) = ((
833:, β, \)
778:, β, /)
760:variety
555:Algebra
441:Algebra
433:Algebra
338:Lattice
329:Lattice
99:improve
6223:
6185:
6154:
6144:
6110:
6093:
6085:
6067:
6035:
5974:β 2, 6
5957:
5934:
5915:
5893:
5876:
5859:
5842:
5396:Monoid
5302:1,411
5167:> 2
5131:where
4894:binary
4865:means
4693:medial
3465:, the
3321:For a
3257:where
2496:Then,
2303:group.
2268:where
2200:Every
2124:β 0 =
2063:, and
2029:Every
1176:)) = (
837:(2, 2)
782:(2, 2)
469:Graded
400:Module
391:Module
290:Domain
209:Monoid
145:is an
29:groups
25:magmas
6198:arXiv
6087:82321
6083:S2CID
6053:(PDF)
6033:S2CID
6015:arXiv
5840:S2CID
5822:arXiv
5746:(PDF)
5569:) β (
5419:Notes
5232:Order
5137:<
4886:unary
4841:,...,
4815:with
4743:(the
4673:) β¦ (
4608:is a
4475:from
4405:is a
4201:when
3291:: an
2747:from
2544:: if
2504:is a
2421:) + (
2393:) = (
2365:) β (
2195:) / 2
2164:field
2110:reals
2031:group
1850:sloop
1846:squag
1639:) = (
1597:) = (
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1128:) β¦ (
1089:pique
1009:Loops
746:, an
723:row.
579:magma
549:image
435:-like
393:-like
331:-like
300:Field
258:-like
232:Magma
200:Group
194:-like
192:Group
151:group
84:, or
51:is a
37:magma
35:is a
6221:ISBN
6183:ISBN
6142:ISBN
6108:ISBN
5955:ISBN
5932:ISBN
5913:ISBN
5891:OCLC
5874:OCLC
5857:OCLC
5726:2024
5577:) =
5565:) β
5557:= ((
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5504:) /
5454:) β
5316:109
5223:OEIS
5213:OEIS
5150:i, j
5146:and
5133:1 β€
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4467:and
4463:Let
4431:) =
3950:and
3761:and
3438:and
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2253:and
2240:) β
2226:and
2158:Any
2131:0 β
2094:The
1965:, β)
1943:) β
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1704:and
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1386:β ((
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1260:and
1248:) β
1240:β ((
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1228:) β
1200:and
1109:, β)
1101:, +)
1076:and
1074:left
1040:and
1015:loop
1003:, β)
979:, β)
927:) /
910:) β
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726:The
695:and
677:and
652:and
601:and
589:and
568:, β)
544:The
265:Ring
256:Ring
174:loop
161:and
141:, a
49:loop
47:. A
31:: A
27:and
6134:doi
6091:Zbl
6075:doi
6025:doi
6011:308
5988:doi
5984:307
5832:doi
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5581:β (
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5516:β (
5496:= (
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4751:or
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4581:in
4503:to
4479:to
4407:map
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3885:if
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3696:if
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3333:of
2759:in
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2282:not
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572:set
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