Algebraic Structures and Variations: From Latin Squares to Lie Quasigroups (original) (raw)
Small latin squares, quasigroups, and loops
Journal of Combinatorial Designs, 2007
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. The best previous results were for Latin squares of order 8 (Kolesova, , quasigroups of order 6 (Bower, 2000) and loops of order 7 . The loops of order 8 have been independently found by " QSCGZ" and Guérin (unpublished, 2001).
2011
In this paper we dene linear over a group and an abelian group binary invertible algebras and characterize the class of such algebras by second-order formulae, namely the ∀∃(∀)-identities. 1. Introduction A quasigroup, (Q; •), of the form, xy = ϕx + a + ψy, where (Q; +) is a group, ϕ, ψ are automorphisms (antiautomorphisms) of (Q; +), and a is a xed element of Q, is called linear (alinear) quasigroup over the group, (Q; +), [2, 6]. All primitive linear (alinear) quasigroups form a variety [6]. A linear quasigroup over an abelian group is called a T-quasigroup [10]. An important subclass of the T-quasigroups is the class of medial quasigroups. A quasigroup (Q; •) is called medial, if the following identity holds: xy • uv = xu • yv. Any medial quasigroup is a T-quasigroup by Toyoda theorem, [3] [8], with the condition, ϕψ = ψϕ. Medial quasigroups have been studied by many authors, namely R.H. Bruck [8], T. Kepka, P. Nemec and J. Ježek [9]-[11], D.S. Murdoch [16], A.B. Romanowska and J.D.H. Smith [17], K. Toyoda [21] and others and this class plays a special role in the theory of quasigroups. T-quasigroups were introduced by T. Kepka and P. Nemec [10, 11]. Later G.B. Belyavskaya characterized the class of T-quasigroups by a system of two identities [5, 7]. A binary algebra (Q; Σ) is called invertible, if (Q; A) is a quasigroup for any operation, A ∈ Σ. The invertible algebras rst were considered by 2010 Mathematics Subject Classication: 20N05 Keywords: quasigroup, invertible algebra, linear algebra, second-order formula, invertible T-algebra, hyperidentity.
2012
In this paper we dene linear over a group and an abelian group binary invertible algebras and characterize the class of such algebras by second-order formulae, namely the ∀∃(∀)-identities. 1. Introduction A quasigroup, (Q; •), of the form, xy = ϕx + a + ψy, where (Q; +) is a group, ϕ, ψ are automorphisms (antiautomorphisms) of (Q; +), and a is a xed element of Q, is called linear (alinear) quasigroup over the group, (Q; +), [2, 6]. All primitive linear (alinear) quasigroups form a variety [6]. A linear quasigroup over an abelian group is called a T-quasigroup [10]. An important subclass of the T-quasigroups is the class of medial quasigroups. A quasigroup (Q; •) is called medial, if the following identity holds: xy • uv = xu • yv. Any medial quasigroup is a T-quasigroup by Toyoda theorem, [3] [8], with the condition, ϕψ = ψϕ. Medial quasigroups have been studied by many authors, namely R.H. Bruck [8], T. Kepka, P. Nemec and J. Ježek [9]-[11], D.S. Murdoch [16], A.B. Romanowska and J.D.H. Smith [17], K. Toyoda [21] and others and this class plays a special role in the theory of quasigroups. T-quasigroups were introduced by T. Kepka and P. Nemec [10, 11]. Later G.B. Belyavskaya characterized the class of T-quasigroups by a system of two identities [5, 7]. A binary algebra (Q; Σ) is called invertible, if (Q; A) is a quasigroup for any operation, A ∈ Σ. The invertible algebras rst were considered by 2010 Mathematics Subject Classication: 20N05 Keywords: quasigroup, invertible algebra, linear algebra, second-order formula, invertible T-algebra, hyperidentity.
The Structure of Idempotent Translatable Quasigroups
Bulletin of the Malaysian Mathematical Sciences Society
We prove the main result that a groupoid of order n is an idempotent and k-translatable quasigroup if and only if its multiplication is given by x • y = (ax + by)(mod n), where a + b = 1(mod n), a + bk = 0(mod n) and (k, n) = 1. We describe the structure of various types of idempotent k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent k-translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical. Keywords Quasigroup • Quadratical quasigroup • k-translatability Mathematics Subject Classification 20M15 • 20N02 2 Preliminary Definitions and Results Recall that a groupoid (Q, •) has property A if it satisfies the identity x y • x = zx • yz [7,18]. It is called right solvable (left solvable) if for any {a, b} ⊆ Q there exists a unique x ∈ G such that ax = b (xa = b). It is left (right) cancellative if x y = xz implies y = z (yx = zx implies y = z). It is a quasigroup if it is left and right solvable. Note that right solvable groupoids are left cancellative, left solvable groupoids are right cancellative, and quasigroups are cancellative. Volenec [18] defined quadratical groupoids as right solvable groupoids satisfying property A and proved some basic properties of these groupoids. Below, we list several
Latin squares, partial latin squares and its generalized quotients
2003
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square.
The Universality of the variety of quasigroups
Journal of the Australian Mathematical Society, 1976
The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra A,/ o ,/,,/ 2 , • • •) where f o ,fi,fi, • • • is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup (B, • > and polynomial operations F o , F,, F 2 , • • • over (B, •) such that (A, /",/,,•••) is a subalgebra of (B, F o , F,, • • •) satisfying exactly the same identities. Moreover, if there are only finitely many / 0 ,/i, • • •, then (B, •) can be taken so that its identities are recursive in those of (A,f o ,f,, • • •). If (A,f o ,f,, • ••> is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.
On autotopies and automorphisms of n-ary linear quasigroups
2004
In this article we study structure of autotopies, automorphisms, autotopy groups and automorphism groups of n- ary linear quasigroups. We find a connection between automorphism groups of some special kinds of n-ary quasigroups (idempotent quasigroups, loops) and some isotopes of these quasigroups. In binary case we find more detailed connections between automorphism group of a loop and automorphism group of some its isotope. We prove that every finite medial n-ary quasigroup of order greater than 2 has a non- identity automorphism group. We apply obtained results to give some information on auto- morphism groups of n-ary quasigroups that correspond to the ISSN code, the EAN code and the UPC code.
Infinitesimal and B∞-algebras, finite spaces, and quasi-symmetric functions
Journal of Pure and Applied Algebra, 2015
Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a B∞-algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.