Volume 44, Issue 3

We present an upper bound for the cardinality of any n -generated algebra in a locally finite variety 𝒱 of algebras. This upper bound depends only on some fundamental numerical invariants of the n -generated subdirectly irreducible algebras in 𝒱 . A theorem characterizing those varieties that contain algebras whose cardinalities achieve the upper bound is proved. Several explicit methods for computing the exact values of these invariants are described. The final section contains detailed concrete examples illustrating applications of the characterization theorem and of the various methods for computing the upper bound.

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.

Let ( A, f ) be a monounary algebra. We describe all monounary algebras ( A, g ) having the same set of quasiorders, Quord( A, f ) = Quord( A, g ). It is proved that if Quord( A, f ) does not coincide with the set of all reflexive and transitive relations on the set A and ( A, f ) contains no cycle with more than two elements, then f is uniquely determined by means of Quord( A, f ). In the opposite case, Quord( A, f ) = Quord( A, g ) if and only if Con( A, f ) = Con( A, g ). Further, we show that, except the case when Quord( A, f ) coincides with the set of all reflexive and transitive relations, if the monounary algebras ( A, f ) and ( A, g ) have the same quasiorders, then they have the same retracts. Next we characterize monounary algebras which are determined by their sets of retracts and connected monounary algebras which are determined by their sets of quasiorders.

This is a survey paper about an algebraic approach to imprecise probabilities. In the first part of it, we outline the work by Walley on imprecise probabilities and the more algebraic approach of Fedel et al.. Then, in the second part we will present some work in progress about a general treatment of upper and lower probabilities over many-valued events and of upper and lower previsions of gambles, by means of Universal Algebra.

In this paper the connection between the right (left) invertible medial algebras and the right (left) group of binary operations is studied. As a consequence, we prove the structure results for transitive modes.

By recent results of M. Stronkowski, it is known that not all modes embed as subreducts into semimodules over commutative unital semirings. Related to this problem is the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provide a general construction of such semirings, along with basic examples and some general properties. The second part of the paper will deal with applications of the general construction to some selected varieties of modes, and will provide a description of semirings determining varieties of semimodules having algebras from these varieties as idempotent subreducts.

The main aim of this paper is to describe the free objects in arbitrary varieties of modals (semilattice ordered idempotent and entropic algebras) and give some new representations of modals.

The paper discusses a representation of modes (idempotent and entropic algebras) as subalgebras of so-called functorial sums of cancellative algebras. We show that each mode that has a homomorphism onto an algebra satisfying a certain additional condition, with corresponding cancellative congruence classes, embeds into a functorial sum of cancellative algebras. We also discuss typical applications of this result.

Modes are idempotent and entropic algebras. Modals are both join semilattices and modes, where the mode structure distributes over the join. Barycentric algebras are equipped with binary operations from the open unit interval, satisfying idempotence, skew-commutativity, and skew-associativity. The article aims to give a brief survey of these structures and some of their applications. Special attention is devoted to hierarchical statistical mechanics and the modeling of complex systems. An additivity theorem for the entropy of independent combinations of systems is proved.

Motivated by the recent study of several researchers on extended-order algebras , introduced by C. Guido and P. Toto as a possible common framework for the majority of algebraic structures used inmany-valued mathematics, the paper focuses on the properties of homomorphisms of the new structures, considering extended-order algebras as a generalization of partially ordered sets. The manuscript also introduces the notion of extended-relation algebra providing a new framework for developing the theory of rough sets.

In this paper a restricted version of the Galois connection between polymorphisms and invariants, called Pol − C Inv, is studied, where the invariant relations are restricted to so-called clausal relations. The lattice of all clones arising from this Galois connection, so-called C -clones, is investigated up to equality of their unary parts, denominated C -monoids. All atoms and co-atoms in the lattice of all C -monoids are characterized.

We show that the system of axioms of a dimonoid is independent and prove that every dimonoid with a commutative operation is a semilattice of archimedean subdimonoids, every dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, every dimonoid with a commutative operation is a semilattice of a -connected subdimonoids and every idempotent dimonoid is a semilattice of rectangular subdimonoids.