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1
Content available remote 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.
2
Content available Modyfications of Csákány's Theorem
100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2000
|
tom 20
|
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.
3
Content available 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.
4
Content available Note on algebraic interior systems
100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2005
|
tom 25
|
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.
5
Content available 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.
6
Content available Implication algebras
100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2006
|
tom 26
|
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.
7
Content available 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.
8
Content available Horizontal sums of basic algebras
100%
Chajda I.
Discussiones Mathematicae - General Algebra and Applications
|
2009
|
tom 29
|
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.
9
Content available 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.
10
Content available Sublattices corresponding to very true operators in commutative basic algebras
63%
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.
11
Content available remote Residuation in orthomodular lattices
63%
Chajda I. , Länger H.
Topological Algebra and its Applications
|
2017
|
tom 5
|
nr 1
1-5
EN
We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.
12
Content available Some modifications of congruence permutability and dually congruence regular varietie
63%
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].
13
Content available Hypersubstitutions in orthomodular lattices
63%
Chajda I. , Länger H.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 1
83-92
EN
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.
14
Content available Relatively complemented ordered sets
63%
Chajda I. , Morávková Z.
Discussiones Mathematicae - General Algebra and Applications
|
2000
|
tom 20
|
nr 2
207-217
EN
We investigate conditions for the existence of relative complements in ordered sets. For relatively complemented ordered sets with 0 we show that each element b ≠ 0 is the least one of the set of all upper bounds of all atoms contained in b.
15
Content available Congruence submodularity
63%
Chajda I. , Halaš R.
Discussiones Mathematicae - General Algebra and Applications
|
2002
|
tom 22
|
nr 2
131-139
EN
We present a countable infinite chain of conditions which are essentially weaker then congruence modularity (with exception of first two). For varieties of algebras, the third of these conditions, the so called 4-submodularity, is equivalent to congruence modularity. This is not true for single algebras in general. These conditions are characterized by Maltsev type conditions.
16
Content available A scheme for congruence semidistributivity
63%
Chajda I. , Horváth E.
Discussiones Mathematicae - General Algebra and Applications
|
2003
|
tom 23
|
nr 1
13-18
EN
A diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the ∧-semidistributive law.
17
Content available Congruence classes in Brouwerian semilattices
63%
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.
18
Content available Ring-like operations is pseudocomplemented semilattices
63%
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.
19
Content available Quasi-implication algebras
63%
Chajda I. , Dušek K.
Discussiones Mathematicae - General Algebra and Applications
|
2002
|
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.
20
Content available Balanced congruences
63%
Chajda I. , Eigenthaler G.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 1
105-114
EN
Let V be a variety with two distinct nullary operations 0 and 1. An algebra 𝔄 ∈ V is called balanced if for each Φ,Ψ ∈ Con(𝔄), we have [0]Φ = [0]Ψ if and only if [1]Φ = [1]Ψ. The variety V is called balanced if every 𝔄 ∈ V is balanced. In this paper, balanced varieties are characterized by a Mal'cev condition (Theorem 3). Furthermore, some special results are given for varieties of bounded lattices.
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