Commutative Quasigroup Research Articles

Commutative quasigroups and semifields from planar functions

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : ( N μ ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q 1 and Q 2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form ( α , α , γ ) .

Divisor graphs and isotopy invariants of commutative quasigroups

Given an element x of a commutative quasigroup A, the x-divisor graph of A is the graph Γx(A) whose vertices are the elements of A such that distinct vertices a and b are adjacent if and only if ab=x. This paper examines how isotopies act on the set {Γx(A) | x∈A}. It is shown that if |A|<∞ is not a multiple of 4, then this set is an isotopy invariant. Moreover, under certain conditions (for example, when A is isotopic to a group), the commutative quasigroup isotopes of A are completely determined by symmetries of {Γx(A) | x∈A}.

Study of Jordan quasigroups and their construction

ABSTRACTJordan quasigroups are commutative quasigroups satisfying the identity . In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.

On the Multiplicative Groups of Free and Free Commutative Quasigroups

We study the structure of the multiplicative groups for the relatively free algebras of some quasigroup varieties, in particular, for the varieties of all quasigroups, all commutative quasigroups, and all TS-quasigroups. In all these cases, the corresponding multiplicative group is free. The work is dedicated to the glorious memory of Prof. Alexandr Alexandrovich Nechaev, who actively explored the possibility of application of quasigroups in cryptography during recent years.

On Commutative Quasigroup

Commutative quasigroups are quasigroups satisfying \(xy=yx\). We construct an infinite family of commutative quasigroup whose smallest member is of order \(4\).

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph Km(n1,n2,…,nm) with vertex set V and m independent sets G1,G2,…,Gm of n1,n2,…,nm vertices respectively. Let G={G1,G2,…,Gm}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V∖Gi for some Gi∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type gu. In this paper, we show that there exists a (k,λ)-cycle frame of type gu for k∈{4,5,6} if and only if g(u−1)≡0(modk), λg≡0(mod2), u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for k≡0(mod2). In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when k≡1(mod2). We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.

On determinability of idempotent medial commutative quasigroups by their endomorphism semigroups

We extend the result of P. Puusemp (Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76-104) about determinability of finite Abelian groups by their endomorphism semigroups to finite idempotent medial commutative quasigroups.

Completing partial commutative quasigroups constructed from partial Steiner triple systems is NP-complete

Deciding whether an arbitrary partial commutative quasigroup can be completed is known to be NP-complete. Here, we prove that it remains NP-complete even if the partial quasigroup is constructed, in the standard way, from a partial Steiner triple system. This answers a question raised by Rosa in [A. Rosa, On a class of completable partial edge-colourings, Discrete Appl. Math. 35 (1992) 293–299]. To obtain this result, we prove necessary and sufficient conditions for the existence of a partial Steiner triple system of odd order having a leave L such that E ( L ) = E ( G ) where G is any given graph.

The varieties of quasigroups of Bol–Moufang type: An equational reasoning approach

A quasigroup identity is of Bol–Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. ( ( x y ) x ) z = x ( y ( x z ) ) . Many well-known varieties of quasigroups are of Bol–Moufang type. We show that there are exactly 26 such varieties, determine all inclusions between them, and provide all necessary counterexamples. We also determine which of these varieties consist of loops or one-sided loops, and fully describe the varieties of commutative quasigroups of Bol–Moufang type. Some of the proofs are computer-generated.

IRREGULAR QUASIVARIETIES OF COMMUTATIVE BINARY MODES

In an earlier paper the authors described the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. This paper extends the earlier result, and provides a description of the lattice of some irregular quasivarieties of commutative binary modes, i.e. quasivarieties that do not contain semilattices.

Quasivarieties of cancellative commutative binary modes

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

Embedding partial steiner triple systems is NP-complete

It is known that any partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w whenever w⩾4 υ+1, and w≡1, 3 (mod 6); moreover, it is conjectured that the same is true whenever w⩾2 υ+1. By way of contrast, it is proved that deciding whether a partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w for any w⩽2 υ−1 is NP-complete. In so doing, it is proved that deciding whether a partial commutative quasigroup can be completed to a commutative quasigroup is NP-complete.