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( L , ϕ ) -representations of algebras

Andrzej Walendziak (1993)

Archivum Mathematicum

In this paper we introduce the concept of an ( L , ϕ ) -representation of an algebra A which is a common generalization of subdirect, full subdirect and weak direct representation of A . Here we characterize such representations in terms of congruence relations.

A note on cylindric lattices

Ivo Düntsch (1993)

Banach Center Publications

0. Introduction. Besides being of intrinsic interest, cylindric (semi-) lattices arise naturally from the study of dependencies in relational databases; the polynomials on a cylindric semilattice are closely related to the queries obtainable from project-join mappings on a relational database (cf. [D] for references). This note is intended to initiate the study of these structures, and only a few, rather basic results will be given. Some problems at the end will hopefully stimulate further research....

Congruence lattices in varieties with compact intersection property

Filip Krajník, Miroslav Ploščica (2014)

Czechoslovak Mathematical Journal

We say that a variety 𝒱 of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every A 𝒱 is closed under intersection. We investigate the congruence lattices of algebras in locally finite, congruence-distributive CIP varieties and obtain a complete characterization for several types of such varieties. It turns out that our description only depends on subdirectly irreducible algebras in 𝒱 and embeddings between them. We believe that the strategy used here can...

Definability for equational theories of commutative groupoids

Jaroslav Ježek (2012)

Czechoslovak Mathematical Journal

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.

Discriminator varieties of Boolean algebras with residuated operators

Peter Jipsen (1993)

Banach Center Publications

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated...

Homomorphic images of finite subdirectly irreducible unary algebras

Jaroslav Ježek, P. Marković, David Stanovský (2007)

Czechoslovak Mathematical Journal

We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty.

On a certain construction of lattice expansions

Hector Gramaglia (2004)

Mathematica Bohemica

We obtain a simple construction for particular subclasses of several varieties of lattice expansions. The construction allows a unified approach to the characterization of the subdirectly irreducible algebras

On distributive trices

Kiyomitsu Horiuchi, Andreja Tepavčević (2001)

Discussiones Mathematicae - General Algebra and Applications

A triple-semilattice is an algebra with three binary operations, which is a semilattice in respect of each of them. A trice is a triple-semilattice, satisfying so called roundabout absorption laws. In this paper we investigate distributive trices. We prove that the only subdirectly irreducible distributive trices are the trivial one and a two element one. We also discuss finitely generated free distributive trices and prove that a free distributive trice with two generators has 18 elements.

On idempotent modifications of M V -algebras

Ján Jakubík (2007)

Czechoslovak Mathematical Journal

The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an M V -algebra 𝒜 we denote by 𝒜 ' , A and ( 𝒜 ) the idempotent modification, the underlying set or the underlying lattice of 𝒜 , respectively. In the present paper we prove that if 𝒜 is semisimple and ( 𝒜 ) is a chain, then 𝒜 ' is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.

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