| 1. Jānis CĪRULIS, Weak Relative Annihilators in Posets , | 1-12 [Abstract] [PDF] |
| 2. K. DENECKE and N. SARASIT, Products of Tree Languages , | 13-36 [Abstract] [PDF] |
| 3. Wojciech DZIK, Remarks on Projective Unifiers, | 37-46 [Abstract] [PDF] |
| 4. Anetta GÓRNICKA, Axiomatization of the Sentential Logic Dual to Sobociński's n - Valued Logic , | 47-54 [Abstract] [PDF] |
| 5. Joanna GRYGIEL, Products of Skeletons of Finite Distributive Lattices, | 55-62 [Abstract] [PDF] |
| 6. Adam KOLANY, Changing a Numbering System is Usually Uncomputable, | 63-68 [Abstract] [PDF] |
| 7. Zofia KOSTRZYCKA, On the Family of Logics Determined by Parasol-Frames, | 69-82 [Abstract] [PDF] |
| 8. Miroslaw KURKOWSKI, On Some Time Properties of Untimed Propositional Implication, | 83-90 [Abstract] [PDF] |
| 9. Sergey A. SOLOVYOV, A Note on Nuclei of Quantale Algebras, | 91-112 [Abstract] [PDF] |
1. Jānis CĪRULIS, Weak Relative Annihilators in Posets
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.
2. K. DENECKE and N. SARASIT, Products of Tree Languages
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 z-product. 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, left-zero and right-zero 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.
3. Wojciech DZIK, Remarks on Projective Unifiers
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 non-discriminator 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.
4. Anetta GÓRNICKA, Axiomatization of the Sentential Logic Dual to Sobociński's n - Valued Logic
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 three-valued logic and nonsense logic W.
5. Joanna GRYGIEL, Products of Skeletons of Finite Distributive Lattices
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 H-irreducible lattices.
6. Adam KOLANY, Changing a Numbering System is Usually Uncomputable
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.
7. Zofia KOSTRZYCKA, On the Family of Logics Determined by Parasol-Frames
We consider the family of logics from NEXT(T2) which are determined by the so-called parasol-frames and we answer the question what the cardinality of the family is.
8. Miroslaw KURKOWSKI,
On Some Time Properties of Untimed Propositional Implication
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.
9. Sergey A. SOLOVYOV, A Note on Nuclei of Quantale Algebras
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
semi-algebra.