| 1. Teresa BIEGANSKA and Katarzyna HALKOWSKA, Generics of P-compatible Varieties | 1 [Abstract] [PDF] |
| 2. Luis F. CACERES-DUQUE, Ideal Theories of some Commutative Rings | 9 [Abstract] [PDF] |
| 3. Wojciech DZIK, Unitary Unification of S5 Modal Logic and its Extensions | 19 [Abstract] [PDF] |
| 4. Joanna GRYGIEL, On Gluing of Lattices | 27 [Abstract] [PDF] |
| 5. Jadwiga KNOP, About a Certain Generalization of the Affine Ratio of Three Points and Unharmonic Ratio of Four Points | 33 [Abstract] [PDF] |
| 6. Zofia KOSTRZYCKA and Marek ZAIONC, On the Density of Truth in Dummett's Logic | 43 [Abstract] [PDF] |
| 7. Krystyna MRUCZEK, Subdirectly Irreducible P-compatible Abelian Groups | 57 [Abstract] [PDF] |
| 8. Tomasz POLACIK, Quantified Intuitionistic Propositional Logic and Cantor Space | 65 [Abstract] [PDF] |
| 9. Bozena STARUCH, Derivation from Partial Knowledge in Partial Models | 75 [Abstract] [PDF] |
| 10. Bozena STARUCH and Bogdan STARUCH, Possible Sets of Equations | 85 [Abstract] [PDF] |
1. Teresa BIEGANSKA and Katarzyna HALKOWSKA, Generics of P-compatible Varieties
For every variety V there exists an algebra A generating V by means of direct products,
subalgebras and homomorphic images, i.e. V=HSP(A). Such algebras are called generics
of V. Obviously, the free algebra over V with ω generators is a generic of V. The
aim of this paper is to find finite generics for some P-compatible varieties.
2. Luis F. CACERES-DUQUE, Ideal Theories of some Commutative Rings
Associated with the congruence relations of any algebra U there is a
propositional theory T(U) whose models are exactly the congruences of U.
In particular, given a commutative ring R with 1 it is possible to show that for any
consistent extension T of T(R) there must always be a sentence α such that
(T u {α}) has a unique model, called an atomic ideal of T. We define extensions
of T(R) which are analogous to taking the Cantor-Bendixon derivative in a topological
space and we present some particular examples of atomic ideals of these theories in
some commutative rings.
3. Wojciech DZIK, Unitary Unification of S5 Modal Logic and its Extensions
It is shown that all extensions of S5 modal logic, both in the standard
formalization and in the formalization with strict implication, as well as all
varieties of monadic algebras have unitary unification.
4. Joanna GRYGIEL, On Gluing of Lattices
We compare the operation on lattices given by A. Wronski to the
operation of gluing of bounded lattices according to a skeleton
introduced by Ch. Herrmann. We prove that these operations differ
in many respects. In particular, sum-irreducible lattices with
respect to these operations do not coincide.
5. Jadwiga KNOP, About a Certain Generalization of the Affine Ratio of Three Points
and Unharmonic Ratio of Four Points
The paper refers to a certain generalization of the affine
ratio. As it is known the affine ratio is defined for three
collinear points. The paper gives a definition of the
n-dimensional affine ratio for n+2 points belonging to an n-dimensional affine
subspace. If n=1, then we obtain a definition of the affine ratio of three points. We
also define the two-dimensional unharmonic ratio of five projective points. For proper
points the two-dimensional unharmonic ratio can be expressed in terms of the two-dimensional
affine ratio.
6. Zofia KOSTRZYCKA and Marek ZAIONC, On the Density of Truth in Dummett's Logic
For the given logical calculus we investigate the size of the
fraction of true formulas of a certain length n against the
number of all formulas of this length. We are especially
interested in asymptotic behaviour of this fraction when n tends
to infinity. If the limit of the fraction exists it represents a
number which we call the density of truth for the
investigated logic. In this paper we apply this approach to the
Dummett intermediate linear logic. This
paper shows the exact density of this logic and demonstrates that
it covers a substantial part of classical propositional calculus.
In fact, despite strictly mathematical means used to solve all discussed
problems, this paper may have a philosophical impact on
understanding to what extent the phenomenon of truth is sporadic or
frequent in random mathematics sentences.
7. Krystyna MRUCZEK, Subdirectly Irreducible P-compatible Abelian Groups
Identities of some special structure like regular identities, normal
identities, externally compatible identities
and generalization of the last two: P-compatible identities have
been studied since the sixties. They have been determined by the structure
of terms occurring in them.
8. Tomasz POLACIK, Quantified Intuitionistic Propositional Logic and Cantor Space
We consider propositional quantification in
intuitionistic logic. We prove that, under topological
interpretation over Cantor space, it enjoys surprising and
interesting properties such as maximum property and a kind of
distribution of existential quantifier over conjunction. Moreover,
by pointing to the appropriate examples, we show that the set of
quantified formulas valid in Cantor space strictly contains the
set of formulas provable in the minimal system of intuitionistic
logic with propositional quantification.
9. Bozena STARUCH, Derivation from Partial Knowledge in Partial Models
We consider partial model, i.e. extensions of partial
algebras by predicates, and we define for them the operator of
possibility on sets of first order sentences.
10. Bozena STARUCH and Bogdan STARUCH, Possible Sets of Equations
We investigate equations for a given partial algebra by appealing to the
class of all total algebras it makes part of.