Our treatment is based on the infinitesimal theory of smooth local bol loops elaborated above and the fact that a moufang loop is nothing more than a left and right bol loop simultaneously equivalently, it is a right bol loop with the left inverse property x. Here we are going to present the infinitesimal theory of smooth local moufang loops. The orthogonal array representation of lis the set of n2 ordered triplets i,j, ij 1. Rectangular quasigroups and rectangular loops arxiv. Elements of quasigroup theory and some its applications in code. This paper is concerned with the numbers of latin squares, quasigroups and loops for small n. Added left bol loops and right bol loops of order pq, for odd primes pq. Loop varieties, free loops and free quasigroups are discussed in 9, 10, and 11. Automated theorem proving in quasigroup and loop theory. In loop theory, autotopisms have been useful in the study of speci.
Pdf a study of new concepts in smarandache quasigroups and. If a given class of quasigroups is isotopic to a highly structured class of loops, that usually means that the given class of quasigroups is itself highly structured. Abstract we present the numbers of isotopy classes and main classes of latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. Library moufang extended by all 5 nonassociative moufang loops of order 81. A variety of quasigroups is the class of all quasigroups satisfying some given set of quasigroup identities. Loops such as bol loops, moufang loops, central loops and extra loops are the most popular loops of bolmoufang type whose isotopic invariance have been considered. Small latin squares, quasigroups, and loops mckay 2007. Smooth moufang loops and malcev algebras springerlink. Attainable classes of groupoids, quasigroups, and loops. Automorphismgroup q now works for quasigroups as well. If the quasigroup component is a onesided loop, then we have a \emphright product left, right loop. Our strategy here is to show that a27 implies commutativity.
Here the nucleus of l is a subgroup of l and if l is a moufang loop then its nucleus is a normal subgroup 2, p. Loops and quasigroups emis the european mathematical. A loop is usually defined as a quasigroup with a neutral element 1 6 q satisfying 1 x x and x 1 x. During the last twentyfive years quite remarkable relations between nonas sociative algebra and differential geometry have been discovered in our work. Changed creation of quasigroups and loops by sections. Small latin squares, quasigroups, and loops request pdf. Bialgebra reducts of hopf algebras are quantum loops, while sufficient conditions are given for quantum loop structure to augment to a hopf algebra. He constructed the structure theory of binarylie algebras. Latex, pdf, ps, html, and as an inline help in gap. Pdf characterization of quasigroups and loops semantic. The loops of order 8 have been independently found by \qscgz anderingu unpublished, 2001.
We survey all known results in the area of quasigroup and loop theory to have been. We classify these loops into nonisotopic classes and show also that all quasigroups of order five are isotopic to one of two nonisotopic loops. Strambach, topologicaland analytical quasigroups and loops and related structures. Pdf it is known that the set of conjugates the conjugate set of a binary quasigroup can contain 1, 2, 3 or 6 elements. Quantum quasigroups and quantum loops are selfdual objects providing a general framework for the nonassociative extension of quantum group techniques. Fixed a bug in isotopismloops now returns a correct isotopism.
Questions tagged quasigroups mathematics stack exchange. Kaplan siberian mathematical journal volume 19, pages 424 433 1978 cite this article. The loops package computing with quasigroups and loops in gap. Six of these axioms form a basis for the variety of rectangular quasigroups. Quantum quasigroups and loops are presented within the general context of a symmetric monoidal category. Rado, a class of ternary quasi groups in german, acta math. For an overview of the theory of loops, readers may check 30, 8, 10, 12, 19, 34. Directproduct now works for a list of quasigroups, loops andor groups. A quasigroup homomorphism is just a homotopy for which the three maps are equal. In the combinatorial setting of the category of sets under the cartesian product, quantum loops just correspond to loops, while finite quantum quasigroups comprise quasigroups equipped with an ordered pair of automorphisms 3. A pair of smarandachely isotopic quasigroups and loops of the. The emphasis will be on the connections between groups, composition algebras, combinatorics, and quasigroups.
Also the theory of smooth quasigroups and loops started to find interesting applications in geometry and physics. Pdf identities in quasigroups and loops 1 what are. Introduction sigma series in pure mathematics hardcover january 1, 1990 by hala o pflugfelder author visit amazons hala o pflugfelder page. All standard gap commands for magmas are therefore available for quasigroups and. A quasigroup is a generalization of a group without associative law or identity element. In the recent past hala 1990 made a description on quasigroups and loops. Further various properties are verified on loops, quasigroups, equasigroups when they are compared to groups and subgroups. Since the presence of an identity element is essential, a loop cannot be empty. Added loopsuptoisomorphism, a function that filters a list of loops up to isomorphism. Idempotent total symmetric quasigroups are precisely i. Identities in quasigroups and loops miami university. There are 4262 nonassociative moufang loops of order 64. A subloop h of a loop g is called a normal subloop of g if for all x, y e g, we. In chapter six, by introducing smarandache special mappings in smarandache groupoids, the smaran.
Attainable classes of groupoids, quasigroups, and loops v. Loops are automatically quasigroups, and both groups and quasigroups are automatically magmas. Goldberg, in handbook of differential geometry, 2000. This will enable us to work with \huge loops in pc groups and matrix groups. How and why moufang loops behave like groups 3 nuc l. Groups can be reached in another way from groupoids, namely through quasi groups. Some alternative representations of a latin square can be useful. Characterization of quasigroups and loops arc journals. Pdf a study of new concepts in smarandache quasigroups. A survey of osborn loops, milehigh conference on loops, quasigroups and nonassociative systems, 1999. Find all the books, read about the author, and more. Directproduct now works for a list of quasigroups, loops and or groups. The theory of quasigroups and loops, chapter ix, heldermann verlag, berlin, toappear. The loops of order 8 have been independently found by qscgz.
This is a companion to our lectures gap and loops, to be delivered at the workshops loops 2007, prague, czech republic. In the lectures we introduce the gap package loops, describe its capabilities, and explain in detail how. Such exotic structures of algebra as quasigroups and loops were obtained from purely geometric structures such as affinely connected spaces. A quasigroup is similar to group, but without the requirement of associativity. The reader may skip this section and use it for later reference. Basic references for quasigroup and loop theory are 1, 2, 3, 10. In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. Completely new documentation, prepared with gapdoc. A pair of smarandachely isotopic quasigroups and loops of.
List of problems in loop theory and quasigroup theory wikipedia. The isotopic invariance or universality of types and varieties of quasigroups and. List of problems in loop theory and quasigroup theory. Identities in quasigroups and loops hannah hoganson and marco tapia 7192012 1 what are quasigroups and loops. The latter paper uses a linearalgebraic approach to moufang loop extensions. We then show which varieties of onesided loops of bolmoufang type have left and right inverse properties. A quasigroup is a generalization of a group without associative law or identity. Quasigroups and loops pdf and the numbers of isomorphism classes of quasigroups and loops, up to order 10. Thiel, 1990, quasigroups of order 6 bower, 2000 and loops of order 7.
Combinatorially, a quasigroupq,is a set qequippedwith a binarymultiplication operationdenoted by or simple juxtaposition of the two arguments, in which speci. The best previous results were for latin squares of order 8 kolesova, lam, and thiel, 28, quasigroups of order 6 bower, 7, and loops of order 7 brant and mullen, 8. A quasigroup is a set endowed with a binary operation, such that for any equation of the form x. Basic references for quasigroup and loop theory are 58. Any two of ralt, lalt, flex implies that there is a twosided identity. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Flex plus either a right or a left identity implies that there is a twosided identity. Cycle structure of autotopisms of quasigroups and latin. Loops can also be characterized equationally as quasigroups satisfying the additional axiom x\x yy. To restrict ones interest is then not just a matter of taste, but necessity.
We now know that the library moufang contains all nonassociative moufang loops of order 64 up to isomorphism. Richard hubert bruck 1971 made a survey of binary systems. Introduction sigma series in pure mathematics on free shipping on qualified orders. There are also many results concerning autotopisms of quasigroups and loops, that is, quasigroups with a neutral element. Quasigroups latin squares, loops, and ip loops are some of the widely studied algebraic structures 2. A quasigroup is a set endowed with a binary operation, such that for any equation of the form x y z, where two of these variables are known, the third one is uniquely determined. It also distinguishes the relations between groups, quasigroups, loops and equasigroups. Phillips submitted on 31 mar 2009 v1, last revised 1 apr 2009 this version, v2. Pomaskina 405 proved that on any rdimensional smooth submanifold v.
Phillips and i worked with aschbacher cal tech on one form. The best previous results were for latin squares of order 8 kolesova, lam and thiel, 1990, quasigroups of order 6 bower, 2000 and loops of order 7 brant and mullen, 1985. Computing with quasigroups and loops in gap gap workshop 2007, braunschweig author. Aquasigroupgis a loop if itcontains an element esuch that g e e g g for all g. This manuscript illustrates the significance of quasigroups when compared to general groups and subgroups. We find a system of identities which axiomatizes right product quasigroups, and use this to find axiom systems for right product left, right loops. Compared to the theory of groups, the theory of quasigroups is considerably older, dating back at least to eulers work on orthogonal latin squares.
A study of new concepts in smarandache quasigroups and loops. The best previous results were for latin squares of order 8 kolesoa,v lam and thiel, 1990, quasigroups of order 6 bower, 2000 and loops of order 7 brant and mullen, 1985. The loops package 7 directly from the master xml documentation. Computing with quasigroups and loops log of changes.