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Decompositions in lattices and some representations of algebras

We first investigate the properties of consistence, strongness and semimodularity, each of which may be viewed as a generalization of modularity. Chapters 2 and 3 present the decomposition theory of lattices. Here we characterize modularity in terms of the Kurosh-Ore replacement property. Next, we study c-decompositions of elements in lattices (the notion of c-decomposition is introduced as a generalization of those of join and direct decompositions). We find a common generalization of the Kurosh-Ore...


Andrzej Walendziak — 2015

Discussiones Mathematicae - General Algebra and Applications

The notion of pseudo-BCH-algebras is introduced, and some of their properties are investigated. Conditions for a pseudo-BCH-algebra to be a pseudo-BCI-algebra are given. Ideals and minimal elements in pseudo-BCH-algebras are considered.

On ideals of pseudo-BCH-algebras

Andrzej Walendziak — 2016

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

In this paper we introduce the notion of a disjoint union of pseudo-BCH-algebras and describe ideals in such algebras. We also investigate ideals of direct products of pseudo-BCH-algebras. Moreover, we establish conditions for the set of all minimal elements of a pseudo-BCH-algebra X to be an ideal of X.

On branchwise commutative pseudo-BCH algebras

Andrzej Walendziak — 2017

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Basic properties of branches of pseudo-BCH algebras are described. Next, the concept of a branchwise commutative pseudo-BCH algebra is introduced. Some conditions equivalent to branchwise commutativity are given. It is proved that every branchwise commutative pseudo-BCH algebra is a pseudo-BCI algebra.

Pseudo-BCH Semilattices

Andrzej Walendziak — 2018

Bulletin of the Section of Logic

In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.

On implicative and maximal ideals of BL-algebras

Andrzej Walendziak — 2014

Commentationes Mathematicae

In the theory of MV-algebras, implicative ideals are studied by Hoo and Sessa. In this paper we define and characterize implicative ideals of BL-algebras. We also investigate maximal ideals of BL-algebras and prove that if an ideal is prime and implicative, then it is maximal. Moreover, we show that an ideal is maximal if and only if the quotient BL-algebra is a simple MV-algebra. Finally, we give the homomorphic properties of implicative and maximal ideals.

On normal filters and congruence relations in BE-algebras

Andrzej Walendziak — 2012

Commentationes Mathematicae

In this paper we introduce the notion of a normal filter in BE-algebras (in transitive BE-algebras filters conicide with normal filters). We discuss some relationships between congruence relations and normal filters of a BE-algebra A (if A is commutative, then we show that there is a bijection between congruence relations and filters in A ). Moreover, we give the construction of quotient algebra A/F of A via a normal filter F of A.

Weak congruences of an algebra with the CEP and the WCIP

Andrzej Walendziak — 2002

Czechoslovak Mathematical Journal

Here we consider the weak congruence lattice C W ( A ) of an algebra A with the congruence extension property (the CEP for short) and the weak congruence intersection property (briefly the WCIP). In the first section we give necessary and sufficient conditions for the semimodularity of that lattice. In the second part we characterize algebras whose weak congruences form complemented lattices.

Semimodularity in lower continuous strongly dually atomic lattices

Andrzej Walendziak — 1996

Archivum Mathematicum

For lattices of finite length there are many characterizations of semimodularity (see, for instance, Grätzer [3] and Stern [6]–[8]). The present paper deals with some conditions characterizing semimodularity in lower continuous strongly dually atomic lattices. We give here a generalization of results of paper [7].

( 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.

Subalgebras and homomorphic images of algebras having the CEP and the WCIP

Andrzej Walendziak — 2004

Czechoslovak Mathematical Journal

In the present paper we consider algebras satisfying both the congruence extension property (briefly the CEP) and the weak congruence intersection property (WCIP for short). We prove that subalgebras of such algebras have these properties. We deduce that a lattice has the CEP and the WCIP if and only if it is a two-element chain. We also show that the class of all congruence modular algebras with the WCIP is closed under the formation of homomorphic images.

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