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Hypersatisfaction of formulas in agebraic systems

Klaus Denecke, Dara Phusanga (2009)

Discussiones Mathematicae - General Algebra and Applications

In [2] the theory of hyperidentities and solid varieties was extended to algebraic systems and solid model classes of algebraic systems. The disadvantage of this approach is that it needs the concept of a formula system. In this paper we present a different approach which is based on the concept of a relational clone. The main result is a characterization of solid model classes of algebraic systems. The results will be applied to study the properties of the monoid of all hypersubstitutions of an...

Hypersubstitutions in orthomodular lattices

Ivan Chajda, Helmut Länger (2001)

Discussiones Mathematicae - General Algebra and Applications

It is shown that in the variety of orthomodular lattices every hypersubstitution respecting all absorption laws either leaves the lattice operations unchanged or interchanges join and meet. Further, in a variety of lattices with an involutory antiautomorphism a semigroup generated by three involutory hypersubstitutions is described.

Ideal extensions of graph algebras

Karla Čipková (2006)

Czechoslovak Mathematical Journal

Let 𝒜 and ℬ be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of 𝒜 by ℬ always exists. We describe (up to isomorphism) all such extensions.

Implication algebras

Ivan Chajda (2006)

Discussiones Mathematicae - General Algebra and Applications

We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties....

Implication and equivalential reducts of basic algebras

Ivan Chajda, Miroslav Kolařík, Filip Švrček (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

A term operation implication is introduced in a given basic algebra 𝒜 and properties of the implication reduct of 𝒜 are treated. We characterize such implication basic algebras and get congruence properties of the variety of these algebras. A term operation equivalence is introduced later and properties of this operation are described. It is shown how this operation is related with the induced partial order of 𝒜 and, if this partial order is linear, the algebra 𝒜 can be reconstructed by means of...

Incidence structures of type ( p , n )

František Machala (2003)

Czechoslovak Mathematical Journal

Every incidence structure 𝒥 (understood as a triple of sets ( G , M , I ) , I ⊆ G × M ) admits for every positive integer p an incidence structure 𝒥 p = ( G p , M p , I p ) where G p ( M p ) consists of all independent p -element subsets in G ( M ) and I p is determined by some bijections. In the paper such incidence structures 𝒥 are investigated the 𝒥 p ’s of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets G and M .

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