
Authors:
Jānis CĪRULIS
Title:Weak Relative Annihilators in Posets
Pages:112
File:bibtex
Abstract ( + )
The notion of relative annihilator, applied to meet semilattices by J. C. Varlet and used by him to define certain relativized versions of distributivity and implicativity of a semilattice, is weakened and adapted for arbitrary posets. In terms of such annihilators, the notions of semidistributivity and weak relative pseudocomplementation, usually considered in the context of meet semilattices and lattices, are defined for posets. Necessary and sufficient conditions are given under which a weakly relatively pseudocomplemented poset is sectionally or relatively pseudocomplemented.

Authors:
K. DENECKE and N. SARASIT
Title:Products of Tree Languages
Pages:1336
File:bibtex
Abstract ( + )
Sets of terms of type τ are called tree languages. The tree language product is an important operation defined on sets of tree languages which maps recognizable tree languages to recognizable tree languages. This tree language product can be described by the superposition of sets of terms. Based on the superposition operation we define a binary associative operation. In the theory of tree languages the product of languages is called the zproduct. The aim of this paper is to study properties of the arising semigroups and its subsemigroups. We are especially interested in idempotent and regular elements, Green's relations ℒ and ℛ, in constant, leftzero and rightzero subsemigroups and in rectangular bands. Since the set of all recognizable tree languages of a given type is closed under the language product, the set of all recognizable tree languages forms a subsemigroup of the semigroup of all tree languages which contains the semigroup of all finite tree languages of the given type. Recognizable tree languages can be generated by regular tree grammars. We characterize all idempotent elements in the semigroup of all recognizable tree languages of type τ by properties of regular grammars. The iteration of the language product plays the role of the Kleene*operation in the theory of formal languages and is one of the regular operations.

Authors:
Wojciech DZIK
Title:Remarks on Projective Unifiers
Pages:3746
File:bibtex
Abstract ( + )
A projective unifier for a unifiable formula α in a logic L is a unifier σ for α (i.e. a substitution making α a theorem of L) such that α⊢ _{L} σ(x)↔ x. Using the result of Burris [3] we observe that every discriminator variety has projective unifiers. Several examples of projective unifiers both in discriminator and in nondiscriminator varieties are given. As an application we show that logics with projective unifiers are almost structurally complete, i.e. every admissible rule with unifiable premises is derivable.

Authors:
Anetta GÓRNICKA
Title:Axiomatization of the Sentential Logic Dual to Sobociński's n  Valued Logic
Pages:4754
File:bibtex
Abstract ( + )
In [4] we introduced the sentential calculus dual to Sobociński's n  valued logic. Here we give an axiomatization of the calculus and prove its completeness. The paper is a continuation of our research on dual logics. In [2], [3], [1] we presented, respectively, axiomatic systems for logics dual to classical logic, Łukasiewicz threevalued logic and nonsense logic W.

Authors:
Joanna GRYGIEL
Title:Products of Skeletons of Finite Distributive Lattices
Pages:5561
File:bibtex
Abstract ( + )
We prove that the skeleton of a product of finitely many finite distributive lattices is isomorphic to the product of skeletons of its factors. Thus, it is possible to construct finite distributive lattices with a given directly reducible skeleton by reducing the problem to the skeleton factors. Although not all possible lattices can be obtained this way, we show that it works for the smallest distributive lattice with the skeleton being a product of Hirreducible lattices.

Authors:
Adam KOLANY
Title:Changing a Numbering System is Usually Uncomputable
Pages:6367
File:bibtex
Abstract ( + )
We show that if we change the base of our numbering system, the induced transformation of the expansion of a given (real) number may be not computable.

Authors:
Zofia KOSTRZYCKA
Title:On the Family of Logics Determined by ParasolFrames
Pages:6981
File:bibtex
Abstract ( + )
We consider the family of logics from NEXT(T_{2}) which are determined by the socalled parasolframes and we answer the question what the cardinality of the family is.

Authors:
Miroslaw KURKOWSKI
Title:On Some Time Properties of Untimed Propositional Implication
Pages:8389
File:bibtex
Abstract ( + )
It is well known that temporal logics are widely used in formal specification and verification of concurrent IT systems. Languages of these logics allow expressing many temporal events and behaviours taking place during system execution. Of course, these languages are more complicated than classical propositional logic language. In verification, properly constructed models of systems can be searched due to investigated properties. Many good algorithms for solving these problems have been proposed. However, from the logical point of view, it is interesting how the classical propositional language has to be extended to describe some interesting system properties. In this paper we show how to construct a computational model of some concurrent systems where some temporal aspects can be described by use of ordinary classical implication only. It is interesting, for example, from the point of view of the problem of cryptographic protocols verification.

Authors:
Sergey A. SOLOVYOV
Title:A Note on Nuclei of Quantale Algebras
Pages:91112
File:bibtex
Abstract ( + )
The paper considers the role of quantale algebra nuclei in representation of quotients of quantale algebras, and in factorization of quantale algebra homomorphisms. The set of all nuclei on a given quantale algebra is endowed with the structure of quantale semialgebra.