Wiesław Dziobiak
University of Puerto Rico
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Algebra Universalis | 1990
Janusz Czelakowski; Wiesław Dziobiak
By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.
Algebra Universalis | 1996
Janusz Czelakowski; Wiesław Dziobiak
Let τ be an algebraic type. To each classK of τ-algebras a consequence relation ⊧K defined on the set of τ-equations is assigned. Some weak forms of the deduction theorem for ⊧K and their algebraic counterparts are investigated. The (relative) congruence extension property (CEP) and its variants are discussed.CEP is shown to be equivalent to a parameter-free form of the deduction theorem for the consequence ⊧K.CEP has a strong impact on the structure ofK: for many quasivarietiesK,CEP implies thatK is actually a variety. This phenomenon is thoroughly discussed in Section 5. We also discuss first-order definability of relative principal congruences. This property is equivalent to the fact that the deduction theorem for ⊧K is determined by a finite family of finite sets of equations. The following quasivarietal generalization of McKenzies [26] finite basis theorem is proved:LetK be quasivariety of algebras of finite type in which the principalK-congruences are definable. ThenK is finitely axiomatizable iff either the classKFSI (of all relatively finitely subdirectly irreducible members ofK) or the class KSI (of all relatively subdirectly irreducible members ofK) is strictly elementary.Applications of the theory to Heyting, interior, Sugihara, and Łukasiewicz algebras are provided.
Studia Logica | 1991
Janusz Czelakowski; Wiesław Dziobiak
We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) ⊑ L and p, A(p, q) ⊢sq.
Studia Logica | 2004
M. E. Adams; Kira V. Adaricheva; Wiesław Dziobiak; A. V. Kravchenko
The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.
Algebra Universalis | 2001
M. E. Adams; Wiesław Dziobiak
Abstract. Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G
Studia Logica | 1981
Wiesław Dziobiak
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Algebra Universalis | 1991
Wiesław Dziobiak
K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.¶We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal.
Studia Logica | 2009
Wiesław Dziobiak; A. V. Kravchenko; Piotr J. Wojciechowski
First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them.
Journal of Symbolic Logic | 1999
Janusz Czelakowski; Wiesław Dziobiak
In [21], D. Pigozzi has proved in a non-constructive way that every relatively congruence distributive quasivariety of finite type generated by a finite set of finite algebras is finitely axiomatizable. In this paper we show that the non-constructive parts of Pigozzis argument can be replaced by constructive ones. As a result we obtain a method of constructing a finite set of quasi-equational axioms for each relatively congruence distributive quasivariety generated by a given finite set of finite algebras of finite type. The method can also be applied to finitely generated congruence distributive varieties.
Studia Logica | 1995
M. E. Adams; Wiesław Dziobiak
We present some equivalent conditions for a quasivariety