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On M-operators of q-lattices

Radomír Halaš — 2002

Discussiones Mathematicae - General Algebra and Applications

It is well known that every complete lattice can be considered as a complete lattice of closed sets with respect to appropriate closure operator. The theory of q-lattices as a natural generalization of lattices gives rise to a question whether a similar statement is true in the case of q-lattices. In the paper the so-called M-operators are introduced and it is shown that complete q-lattices are q-lattices of closed sets with respect to M-operators.

Annihilators in BCK-algebras

Radomír Halaš — 2003

Czechoslovak Mathematical Journal

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra 𝒜 . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice 𝒟 ( A ) of all deductive systems on 𝒜 . Moreover, relative annihilators of C 𝒟 ( A ) with respect to B 𝒟 ( A ) are introduced and serve as relative pseudocomplements of C w.r.t. B in 𝒟 ( A ) .

Pseudocomplemented ordered sets

Radomír Halaš — 1993

Archivum Mathematicum

The aim of this paper is to transfer the concept of pseudocomplement from lattices to ordered sets and to prove some basic results holding for pseudocomplemented ordered sets.

Relative polars in ordered sets

Radomír Halaš — 2000

Czechoslovak Mathematical Journal

In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of R -polars are studied. Connections between R -polars and prime ideals, especially in distributive sets, are found.

Distributive implication groupoids

Ivan ChajdaRadomir Halaš — 2007

Open Mathematics

We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.

Congruence submodularity

Ivan ChajdaRadomír Halaš — 2002

Discussiones Mathematicae - General Algebra and Applications

We present a countable infinite chain of conditions which are essentially weaker then congruence modularity (with exception of first two). For varieties of algebras, the third of these conditions, the so called 4-submodularity, is equivalent to congruence modularity. This is not true for single algebras in general. These conditions are characterized by Maltsev type conditions.

Subdirectly irreducible sectionally pseudocomplemented semilattices

Radomír HalašJan Kühr — 2007

Czechoslovak Mathematical Journal

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented...

Varieties satisfying the triangular scheme need not be congruence distributive

Ivan ChajdaRadomír Halaš — 2007

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

A diagrammatic scheme characterizing congruence distributivity of congruence permutable algebras was introduced by the first author in 2001. It is known under the name Triangular Scheme. It is known that every congruence distributive algebra satisfies this scheme and an algebra satisfying the Triangular Scheme which is not congruence distributive was found by E. K. Horváth, G. Czédli and the autor in 2003. On the other hand, it was an open problem if a variety of algebras satisfying the Triangular...

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