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1
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Horizontal sums of basic algebras

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2009
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tom 29
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nr 1
21-33
EN
The variety of basic algebras is closed under formation of horizontal sums. We characterize when a given basic algebra is a horizontal sum of chains, MV-algebras or Boolean algebras.
2
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Distributivity of bounded lattices with sectionally antitone involutions

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2005
|
tom 25
|
nr 2
155-163
EN
We present a simple condition under which a bounded lattice L with sectionally antitone involutions becomes an MV-algebra. In thiscase, L is distributive. However, we get a criterion characterizingdistributivity of L in terms of antitone involutions only.
3
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Commutative directoids with sectional involutions

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2007
|
tom 27
|
nr 1
49-58
EN
The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.
4
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Congruences on semilattices with section antitone involutions

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2010
|
tom 30
|
nr 2
207-215
EN
We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.
5
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Implication algebras

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
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2006
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tom 26
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nr 2
141-153
EN
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.
6
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Modyfications of Csákány's Theorem

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2000
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tom 20
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nr 1
37-41
EN
Varieties whose algebras have no idempotent element were characterized by B. Csákány by the property that no proper subalgebra of an algebra of such a variety is a congruence class. We simplify this result for permutable varieties and we give a local version of the theorem for varieties with nullary operations.
7
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Note on algebraic interior systems

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2005
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tom 25
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nr 2
149-153
EN
We get an interrelation between an algebraic closure system and its conjugated interior system. We introduce the concept of algebraic interior system and we get its representation.
8
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A groupoid characterization of Boolean algebras

100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2004
|
tom 24
|
nr 2
177-184
EN
We present a groupoid which can be converted into a Boolean algebra with respect to term operations. Also conversely, every Boolean algebra can be reached in this way.
9
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Algebraic axiomatization of tense intuitionistic logic

100%
Chajda I.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1185-1191
EN
We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.
10
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Quasi-implication algebras

64%
Chajda I., Dušek K.
Discussiones Mathematicae - General Algebra and Applications
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2002
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tom 22
|
nr 2
183-198
EN
A quasi-implication algebra is introduced as an algebraic counterpart of an implication reduct of propositional logic having non-involutory negation (e.g. intuitionistic logic). We show that every pseudocomplemented semilattice induces a quasi-implication algebra (but not conversely). On the other hand, a more general algebra, a so-called pseudocomplemented q-semilattice is introduced and a mutual correspondence between this algebra and a quasi-implication algebra is shown.
11
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Ring-like operations is pseudocomplemented semilattices

64%
Chajda I., Länger H.
Discussiones Mathematicae - General Algebra and Applications
|
2000
|
tom 20
|
nr 1
87-95
EN
Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.
12
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Remarks on pseudo MV-algebras

64%
Chajda I., Kolařík M.
Discussiones Mathematicae - General Algebra and Applications
|
2009
|
tom 29
|
nr 1
5-19
EN
Pseudo MV-algebras (see e.g., [4, 6, 8]) are non-commutative extension of MV-algebras. We show that every pseudo MV-algebra is isomorphic to the algebra of action functions where the binary operation is function composition, zero is x ∧ y and unit is x. Then we define the so-called difference functions in pseudo MV-algebras and show how a pseudo MV-algebra can be reconstructed by them.
13
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Semilattices with sectional mappings

64%
Chajda I., Eigenthaler G.
Discussiones Mathematicae - General Algebra and Applications
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2007
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tom 27
|
nr 1
11-19
EN
We consider join-semilattices with 1 where for every element p a mapping on the interval [p,1] is defined; these mappings are called sectional mappings and such structures are called semilattices with sectional mappings. We assign to every semilattice with sectional mappings a binary operation which enables us to classify the cases where the sectional mappings are involutions and / or antitone mappings. The paper generalizes results of [3] and [4], and there are also some connections to [1].
14
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Orthorings

64%
Chajda I., Länger H.
Discussiones Mathematicae - General Algebra and Applications
|
2004
|
tom 24
|
nr 1
137-147
EN
Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.
15
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The rings which are Boolean

64%
Chajda I., Švrček F.
Discussiones Mathematicae - General Algebra and Applications
|
2011
|
tom 31
|
nr 2
175-184
EN
We study unitary rings of characteristic 2 satisfying identity $x^p = x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p = 2^{n} - 2$ or $p = 2^{n} - 5$ or $p = 2^{n} + 1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form $2^{q} + 2m + 1$ or $2^{q} + 2m$ where q is a natural number and $m ∈ {1,2,...,2^{q} - 1}$.
16
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A common approach to directoids with an antitone involution and D-quasirings

64%
Chajda I., Kolařík M.
Discussiones Mathematicae - General Algebra and Applications
|
2008
|
tom 28
|
nr 2
139-145
EN
We introduce the so-called DN-algebra whose axiomatic system is a common axiomatization of directoids with an antitone involution and the so-called D-quasiring. It generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebras and Boolean rings.
17
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Some modifications of congruence permutability and dually congruence regular varietie

64%
Chajda I., Eigenthaler G.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 2
165-174
EN
It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already by J. P≥onka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].
18
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Sublattices corresponding to very true operators in commutative basic algebras

64%
Chajda I., Švrček F.
Discussiones Mathematicae - General Algebra and Applications
|
2014
|
tom 34
|
nr 2
183-189
EN
We introduce the concept of very true operator on a commutative basic algebra in a way analogous to that for fuzzy logics. We are motivated by the fact that commutative basic algebras form an algebraic axiomatization of certain non-associative fuzzy logics. We prove that every such operator is fully determined by a certain relatively complete sublattice provided its idempotency is assumed.
19
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Two constructions of De Morgan algebras and De Morgan quasirings

64%
Chajda I., Eigenthaler G.
Discussiones Mathematicae - General Algebra and Applications
|
2009
|
tom 29
|
nr 2
169-180
EN
De Morgan quasirings are connected to De Morgan algebras in the same way as Boolean rings are connected to Boolean algebras. The aim of the paper is to establish a common axiom system for both De Morgan quasirings and De Morgan algebras and to show how an interval of a De Morgan algebra (or De Morgan quasiring) can be viewed as a De Morgan algebra (or De Morgan quasiring, respectively).
20
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Congruence classes in Brouwerian semilattices

64%
Chajda I., Länger H.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 2
229-237
EN
Brouwerian semilattices are meet-semilattices with 1 in which every element a has a relative pseudocomplement with respect to every element b, i. e. a greatest element c with a∧c ≤ b. Properties of classes of reflexive and compatible binary relations, especially of congruences of such algebras are described and an abstract characterization of congruence classes via ideals is obtained.
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