Wiktor Bartol

On the coverings by tolerance classes

A tolerance is a reflexive and symmetric, but not necessarily transitive, binary relation. Contrary to what happens with equivalence relations, when dealing with tolerances one must distinguish between blocks (maximal subsets where the tolerance is a total relation) and classes (the class of an element is the set of those elements tolerable with it). Both blocks and classes of a tolerance on a set define coverings of this set, but not every covering of a set is defined in this way. The characterization of those coverings that are families of blocks of some tolerance has been known for more than a decade now. In this paper we give a characterization of those coverings of a finite set that are families of classes of some tolerance.

Helena Rasiowa, 1917--1994

This is a biogram of Professor Helena Rasiowa (1917–1994) one of the leading representatives of logicians from Warsaw. She was not only the great scientist but also the great human being. Rasiowa influenced numerous researchers from all over the world, especially by her results in algebraic logic, as well as by her great contribution for the mathematical community in all respect. She is also co-founder of the Pawlak-Rasiowa School of Artificial Intelligence (AI) (Jankowski and Skowron, Andrzej Mostowski and Foundational Studies, pp 106–143, 2008).

On the lattice of fuzzy weak subalgebras of a fuzzy partial algebra

Fuzzy partial algebras can be understood as incomplete algebraic structures whose elements are uncertain or imprecise, and thus fuzzy weak subalgebras of a fuzzy algebra correspond to structural approximations of the knowledge represented by the fuzzy algebra. In this paper we characterize those lattices that are isomorphic to the lattice of fuzzy weak subalgebras of a fuzzy partial algebra, and we discuss what information about a fuzzy monounary partial algebra can be extracted from its lattice of fuzzy weak subalgebras.

Syntactical Content of Finite Approximations of Partial Algebras

In this paper we give a syntactical answer to the following question: What do we actually know about a partial algebra when we know its set of weak or relative subalgebras with cardinal smaller than a fixed bound, if we do not have any information on how they are linked to each other within the algebra?

Mono-unary algebras are uniquely determined by their lattices of fuzzy weak subalgebras☆

In this paper we give an abstract characterization of the lattices that are isomorphic to the lattice of fuzzy weak subalgebras of some partial algebra, and we show that from this lattice we can extract more information about the algebra than from its lattice of weak subalgebras. We use it to prove that the directed graph associated to a unary partial algebra is always uniquely determined (up to isomorphisms) by the algebras lattice of fuzzy weak subalgebras, and in particular that if two mono-unary partial algebras (i.e., two partial algebras over a signature containing only one operation symbol, which is moreover unary) have their lattices of fuzzy weak subalgebras isomorphic, then they are isomorphic.