Algebra Universalis最新文献

The aim of this article is to develop a Priestley-style duality for the variety of DN-algebras. In order to achieve this, we use the concept of free distributive lattice extension of a DN-algebra. We establish a connection with the Priestley duality for distributive lattices. Finally, we present topological descriptions for the lattice of filters, for the lattice of congruences, and for certain kinds of subalgebras of a DN-algebra.

The present paper deals with (L_0)-valued measures and the associated Orlicz–Kantorovich lattice. The considered Orlicz–Kantorovich lattice, endowed with the Luxemburg norm, is represented as a measurable bundle of classical Orlicz spaces associated with scalar measures. This kind of representation allows us to investigate positive contractions and apply the corresponding zero-two laws on the classical Orlicz spaces, to prove vector versions of “zero-two” laws on the considered Orlicz–Kantorovich space.

Orrin Frink in 1962 stated incorrectly, that there exist free pseudocomplemented meet-semilattices, which are not lattices. This is disproved in the present paper. We have shown that any free PCS is in fact a sectionally pseudocomplemented lattice.

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all ((C_1,C_2))-clonoids of Boolean functions are determined for an arbitrary clone (C_1) and for any clone (C_2) containing the clone of self-dual monotone functions.

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, (texttt{DNA})-logics and dependence logic.

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Słupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4 have this property.

We develop a common semantic framework for the interpretation both of ({textbf {IPC}}), the intuitionistic propositional calculus, and of logics weaker than ({textbf {IPC}}) (substructural and subintuitionistic logics). To this end, we prove a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the full subcategory of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.

From work of Vaughan-Lee in [12] it follows that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan-Lee gives an explicit example of a nilpotent loop of order 12 that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.

As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity. In particular, it is shown that an algebraic closure system with arity n in which the empty set is closed and every finitely generated closed set is countable can be realized by a single ((n+1))-ary operation. The algebraic closure system of cosets on any group is realized by the single ternary Mal’cev term (xy^{-1}z). It is shown that the closure system of cosets on an Abelian group A can be realized by a single binary operation if and only if A has at most one element of order 2. Similar results are obtained for modules over an arbitrary ring.

Let ({mathcal {V}}) be a variety of algebras of some type (Omega ). An interest to describing automorphisms of the category (Theta ^0 ({mathcal {V}})) of finitely generated free ({mathcal {V}})-algebras was inspired by development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and applied by B. Plotkin and the author. The method is to find all terms in the language of a given variety which determine such (Omega )-algebras that are isomorphic to a given (Theta ^0 ({mathcal {V}}))-algebra and have the same underlying set with it. But this method can be applied only to automorphisms which take all objects to isomorphic ones. The aim of the present paper is to suggest another method which works in more general setting. This method is based on two main theorems. The first of them gives a general description of automorphisms of categories which are supplied with a faithful representative functor into the category of sets. The second one shows how to obtain the full description of automorphisms of the category (Theta ^0 ({mathcal {V}})). This part of the paper ends with two examples. The first of them shows the preference of our method in a known situation (the variety of all semigroups) and the second one demonstrates obtaining new results (the variety of all modules over arbitrary ring with unit). The last section contains some applications to universal algebraic geometry.