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We investigate {left, right, middle} units of quasigroups and families of identities which might imply their existence. A prominent role is played by the newly introduced notion of derivative operation, generalizing Belousov's notions of left/right derivative operations for quasigroups. Partial solutions of the Belousov's Problem # 18 and its generalizations are obtained. Several related problems remain open.

We deal with solutions of classical linear equations a · x = b and y · a = b , applying a particular lattice valued fuzzy technique. Our framework is a structure with a binary operation · (a groupoid), equipped with a fuzzy equality. We call it a fuzzy quasigroup if the above equations have unique solutions with respect to the fuzzy equality. We prove that a fuzzy quasigroup can equivalently be characterized as a structure whose quotients of cut-substructures with respect to cuts of the fuzzy equality are classical quasigroups . Analyzing two approaches to quasigroups in a fuzzy framework, we prove their equivalence. In addition, we prove that a fuzzy loop (quasigroup with a unit element) which is a fuzzy semigroup is a fuzzy group and vice versa. Finally, using properties of these fuzzy quasigroups, we give answers to existence of solutions of the mentioned linear equations with respect to a fuzzy equality, and we describe solving procedures. Quasigroups and other related structures are an algebraic tool successfully applied up to now in coding theory and cryptology. In our work we propose related applications.

According to S. Krstic, there are only four quadratic varieties which are closed under isotopy. We give a simple procedure generating quadratic identities and deciding which of the four varieties they define. There are about 37000 such identities with up to five variables.

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen 4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych. 27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

Krstic initiated the use of cubic graphs in solving quasigroup equations. Based on his work, Krapež and Živkovic proved that there is a bijective correspondence between classes of parastrophically equivalent parastrophically uncancellable generalized quadratic functional equations on quasigroups and three-connected cubic (multi)graphs. We use the list of such graphs given in the literature to verify existing results on equations with three, four and five variables and to prove new results for equations with six variables. We start with 14 nonisomorphic graphs with ten vertices, choose a set of 14 representative parastrophically nonequivalent equations and give their general solutions. A case of equations with seven and more variables is briefly discussed. The problem of Sokhats’kyi concerning a property which distinguishes visually two parastrophically nonequivalent equations with four variables is solved.

We define a special class of linear quasigroup functional equations and call them height preserving equations (short for the equations which preserve the height of variables). It is proved that a quasigroup satisfying a height preserving but not Belousov equation is isotopic to an abelian group. The formulas of a general solution are also given. Some results of Belousov are discussed and partially generalized.

We define almost quasigroups, a new class of groupoids which generalize quasigroups, and prove several representation theorems for them, essentially reducing them to loops (see Theorems 1, 2 and 9). Some well-known theorems on quasigroups are generalized, notably the theorems of A. A. Albert (Theorems 8, 9 and 10).¶We also define the normal form of equations and show that every generalized linear functional equation Eq on almost quasigroups is equivalent to a system consisting of several equations with at most one variable each, and one equation in the normal form, with the same number of variables as Eq. Eventually, the general solution of the generalized linear functional equations on almost quasigroups with at most two variables is given.¶We plan to solve other generalized linear functional equations in subsequent papers.

The use of quasigroups in cryptography is increasingly popular. One method to find quasigroups suitable for cryptographic purposes is to use identity sieves, i.e. to find appropriate identities and check candidate quasigroups against them. We propose the use of functional equation approach to this problem. Namely, every identity can be considered as a functional equation and solutions to these equations as models of given identities. The identity i.e. functional equation can be transformed into related generalized functional equation which is suitable for algebraic treatment. A new method of solution of quadratic and parastrophically uncancelablle equations is given, using trees and dichotomies (a special equivalence relations). General solution is given by closed formulas. The quasigroups obtained can be further filtered using much simpler conditions.

V. D. Belousov (1925–88) made numerous contributions to the study of quasigroups. In particular, his lengthy 1966 paper “Balanced identities in quasigroups” [4] contains what has been described as a “very significant” and “remarkable” theorem [11, pp. 68–69]. Remarkable though it was, this theorem provided only a partial answer to the question as to which balanced equations on quasigroups gave rise to group isotopes. Although not specifically addressed in the paper [12], a characterization of the balanced equations in question may be derived from a generalization of Belousov's Theorem due to E. Falconer. The first author explicitly solved the problem in 1979; however his characterization was of a technical nature and depended on machinery developed over three papers [13]. In 1985 Belousov found a characterization which is not only elegant but also lends itself to a simple proof [5]. The purpose of this paper is to provide sufficient background for the non specialist to understand and enjoy what we too would describe as “a remarkable theorem”.

A type of groupoid called a rectangular quasigroup is defined as a direct product of a left zero semigroup, a quasigroup and a right zero semigroup. We give three different axiom systems for these groupoids. Some important properties of rectangular quasigroups are derived, the solvability of the word problem among them.