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Decompositions in lattices and some representations of algebras

100%
Walendziak A.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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 Theorem and the Schmidt-Ore Theorem for arbitrary modular lattices, solving a problem of G. Grätzer. The decomposition theory for lattices enables us to develop a structure theory for algebras. Some kinds of representations of algebras are studied. We consider weak direct representations of a universal algebra. The existence of such representations is considered. Finally, we introduce the notion of an ⟨ℒ,ψ⟩-product of algebras. We give sufficient conditions for an algebra to be isomorphic to an ⟨ℒ,ψ⟩-product with simple factors and with directly indecomposable factors. Some applications to subdirect, full subdirect and weak direct products are indicated.
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On maximal ideals of pseudo-BCK-algebras

95%
Walendziak A.
Discussiones Mathematicae - General Algebra and Applications
|
2011
|
tom 31
|
nr 1
61-73
EN
We investigate maximal ideals of pseudo-BCK-algebras and give some characterizations of them.
3
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Pseudo-BCH-algebras

95%
Walendziak A.
Discussiones Mathematicae - General Algebra and Applications
|
2015
|
tom 35
|
nr 1
5-19
EN
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.
4
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Pseudo-BCH Semilattices

95%
Walendziak A.
Bulletin of the Section of Logic
|
2018
|
tom 47
|
nr 2
EN
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.
5
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On branchwise commutative pseudo-BCH algebras

95%
Walendziak A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2017
|
tom 71
|
nr 2
EN
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.
6
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Strong ideals and horizontal ideals in pseudo-BCH-algebras

95%
Walendziak A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2016
|
tom 15
EN
In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.
7
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On implicative and maximal ideals of BL-algebras

95%
Walendziak A.
Commentationes Mathematicae
|
2014
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tom 54
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nr 2
EN
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.
8
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On normal filters and congruence relations in BE-algebras

95%
Walendziak A.
Commentationes Mathematicae
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2012
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tom 52
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nr 2
EN
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.
9
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The p-semisimple property for some generalizations of BCI algebras and its applications

60%
Obojska L., Walendziak A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2020
|
tom 19
79-94
EN
This paper presents some generalizations of BCI algebras (the RM, tRM, *RM, RM**, *RM**, aRM**, *aRM**, BCH**, BZ, pre-BZ and pre-BCI algebras). We investigate the p-semisimple property for algebras mentioned above; give some examples and display various conditions equivalent to p-semisimplicity. Finally, we present a model of mereology without antisymmetry (NAM) which could represent a tRM algebra.
10
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Some characterizations of pseudo-BL-chains

60%
Walendziak A., Wojciechowska-Rysiawa M.
Commentationes Mathematicae
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2011
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tom 51
|
nr 2
EN
Pseudo-BL-chains are linearly ordered pseudo-BL-algebras. Characterizations of them in terms of concepts of lattice theory are given.
11
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On maximal ideals of pseudo \(MV\)-algebras

60%
Dymek G., Walendziak A.
Commentationes Mathematicae
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2007
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tom 47
|
nr 1
EN
We investigate maximal ideals of pseudo MV-algebras and give some characterizations of them. Some properties of a family of maximal ideals of a pseudo MV-algebra generating this algebra are shown as well. Finally, we are interested in finding an example of a pseudo MV-algebra generated by its maximal ideal.
12
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Some results on pseudo-Q algebras

49%
Rezaei A., Saeid A. B., Walendziak A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2017
|
tom 16
EN
The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.
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