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In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDP n , n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP 1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP 1, LV (RDP 1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP 1 and conclude that LV (RDP 1) is countably infinite. An equational basis for each proper subvariety of RDP 1 is given. To study the subvarieties RDP n with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2 ℵ 0 . We also use Priestley duality to prove that RDP and each of its subvarieties RDP n are generated by their finite members.

An n-ary local polymorphism of a given monounary algebra A is a homomorphism from a finitely generated subalgebra of An to A. A is n-polymorphism-homogeneous if each n-ary local polymorphism can be extended to a global polymorphism. Then A is called polymorphism-homogeneous, if it is n-polymorphism-homogeneous for each positive integer n. In this paper we describe all polymorphism-homogeneous monounary algebras.

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.

The notion of idempotent modification of an algebra was introduced by Ježek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.

The notion of internal subdirect decomposition can be defined in each variety of algebras. In the present note we prove the validity of a cancellation rule concerning such decompositions for lattice ordered groups and for GMV-algebras. For the case of groups, this cancellation rule fails to be valid.

We consider here some notions and results about sets of generators of finite algebras motivated by the case of finite groups. We illustrate these notions by simple examples closed to lattices and semigroups. Next, we examine these notions in the case of groups with operators.

A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.

A permutational automaton is a fibered automaton with one-element event set or an empty state space. This paper characterizes all subdirectly irreducible permutational automata.

This is the continuation of the paper “Transformations between Menger systems”. To define when two universal algebras with bases “are the same”, here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of theHence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformationsUniversal transformations allow us to check theContrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

In this paper we introduce the notion of a

We determine the minimally irreducible groups with proper non-soluble socle, whose degree is the product of two primes.