Narrow Results

The notion of binary representation of algebras with at most two binary operations is introduced in this paper, and the binary version of Cayley theorem for distributive lattices is given by hyperidentities. In particular, we get the binary version of Cayley theorem for DeMorgan and Boolean algebras.

P-Compatibility is a hereditary property of identities which generalizes the properties of normality and externality of identities. Chajda characterized the normalization of a variety by an algebraic construction called a choice algebra. In this paper, we generalize this characterization to the least P-compatible variety P(V) determined by a variety V for any partition P using P-choice algebras. We also study the clone of (strongly) P-compatible n-ary terms of a variety V, and relate identities of this clone to (strongly) P-compatible hyperidentities of the variety V.

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. Debrecen4 (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.

In this paper, using the second-order formulas, we obtained characterizations of invertible algebras principally isotopic to a group or an abelian group.

In this paper, the Belousov theorem on linearity of invertible algebras with the Schauffler $\forall \exists (\forall )$-identity is extended over the other $\forall \exists (\forall )$-identities of associativity. As a consequence, we obtain the equivalency of the considered $\forall \exists (\forall )$-identities of associativity and nontrivial hyperidentities of associativity in systems of groups. For the considered formulas, we prove the Schauffler-type theorems, too.

A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.

In [5] the concept of an ID-semiring (S;+,.) as an algebra of type (2,2) where (S;+) and (S;.) are idempotent semigroups (bands) and where four distributive laws: xy + z ≈ (x + z)(y + z), x + yz ≈ (x + y)(x + z), x(y + z) ≈ xy + xz, (x + y)z ≈ xz + xz are satisfied, was introduced. An ID-semiring (S; +,.) is normal if x + y + u + v ≈ x + u + y + v and xyuv ≈ xuyv are identities. In this paper we want to determine the least and the greatest solid variety of normal ID-semirings, i.e. varieties in which every identity is satisfied as a hyperidentity. The result is that the variety of all normal ID-semirings is solid. The variety RA_(2,2) of rectangular algebras of type (2,2), i.e. the variety generated by all projection algebras of type (2,2) is the least non-trivial solid variety of normal ID-semirings.

Partial operations occur in the algebraic description of partial recursive functions and Turing machines (cf. A. I. Mal'cev⁸). Similarly to total operations superposition operations can also be defined on sets of partial operations. A clone of partial operations is a set of partial operations defined on the same base set A which is closed under superposition and contains all total projections. The collection of all clones of partial operations defined on a set A forms a complete lattice. For a finite nonempty set A this lattice is atomic and dually atomic. A partial algebra is said to be strongly solid if every strong identity of is satisfied as a strong hyperidentity in , i.e. if it is satisfied after any replacement of operation symbols by derived term operations of of the corresponding arity. A clone C of partial operations is called strongly solidifyable if there is a partial algebra such that C is equal to the clone of all term operations of . In this paper we determine all minimal strongly solidifyable clones of partial operations defined on a finite nonempty set A.