Let T be a topological space and let A be a subset of T . Recall that A is said to be a closed domain of T if A = IntA and A is said to be an open domain of T if A = IntA (see e.g. [8], [14]). Some simple generalization of these notions is the following one. A is said to be a domain of T provided IntA ⊆ A ⊆ IntA (see [14] and compare [7]). In this paper certain connections between these concept...

We give a quadratic algorithm for the following structure identification problem: given a Boolean relation R and a finite set S of Boolean relations, can the relation R be expressed as a conjunctive query over the relations in the set S? Our algorithm is derived by first introducing the concept of a plain basis for a co-clone and then identifying natural plain bases for every co-clone in Post’s...

In the present report, we study collections of contact relations on a fixed Boolean algebra and show that they can be provided with a rich lattice structure. This follows from a representation theorem which associates with each contact relation a closed graph on the ultrafilter space of the underlying Boolean algebra and vice versa. We also consider collections of special contact relations whic...

Let T be a topological space and let A be a subset of T . Recall that A is said to be a closed domain of T if A = IntA and A is said to be an open domain of T if A = IntA (see e.g. [8], [15]). Some simple generalization of these notions is the following one. A is said to be a domain of T provided IntA ⊆ A ⊆ IntA (see [15] and compare [7]). In this paper certain connections between these concept...

It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and...

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into Heyting algebra topos theory. While distributivity does not always hold lattices, it algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, condition that any element transformed through corresponds to original lattice. show these equivalent, that, only ca...

We study the complexity of network dynamics in a couple of very different model classes: The traditional random Boolean networks (RBN) and Frisch-Hasslacher-Pomeau lattice gas automata (FHP). For this we formulate the FHP dynamics as a probabilistic Boolean network (PBN). We use the set complexity of successive network states to assess the complexity of the dynamics. We find that the complexity...

In 1897, Dedekind [6] posed the problem of estimating the number of antichains in the Boolean lattice; in particular, he asked whether the logarithm of the number is asymptotic to the size of the middle layer of the n-dimensional Boolean lattice Bn. Although Kleitman confirmed the truth of this conjecture in 1969 [13], enumerating antichains in Bn has continued to generate interest in the mathe...

At present, the filter theory of $BL$textit{-}algebras has been widelystudied, and some important results have been published (see for examplecite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor $BL$textit{-}algebras. Note that fil...

It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and...