21 papers
Orthogonal adjointness in posets with
Michal Botur, Ivan Chajda, Helmut Länger
Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with . A pair of operators ,…
Lattices and semilattices derived from commutative rings of characteristic 2 satisfying the identity
Ivan Chajda, Miroslav Kolařík, Helmut Länger
We prove that a commutative ring of characteristic satisfying the identity together with the binary relation on defined by…
Residuated mappings and homomorphisms in posets
Ivan Chajda, Helmut Länger
The concept of weakly regular residuated mappings was investigated for lattices recently by S. Radeleczki and L. Veres. We modify this concept for posets. We define so-called opera…
Varieties and quasivarieties of lattices with complementation
V. Cenker, I. Chajda, J. Kühr +1
We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses sat…
A Cayley theorem for posets
Ivan Chajda, Helmut Länger
We show that every poset P=(P,\le) satisfying the Ascending Chain Condition can be isomorphically embedded into the poset of all mappings from P to the set A(P) of all antichains o…
Operators on complemented posets
Michal Botur, Ivan Chajda, Helmut Länger
Given a complemented poset P, we can assign to every element x of P the set x^+ of all its complements. We study properties of the operator ^+ on P, in particular, we are intereste…
Lagrange-like interpolation in unitary rings, Boolean algebras and Boolean posets
Ivan Chajda, Helmut Länger
It is known that every function with a finite support over a given field can be interpolated by means of the Lagrangian polynomial. The question is if a similar interpolation is po…
Properties of the symmetric difference in lattices with complementation
Václav Cenker, Ivan Chajda, Helmut Länger
The symmetric difference in Boolean lattices can be defined in two different but equivalent forms. However, it can be introduced also in every bounded lattice with complementation…
Operators Max L and Min U and duals of Boolean posets
Ivan Chajda, Miroslav Kolařík, Helmut Länger
When working with posets which are not necessarily lattices, one has a lack of lattice operations which causes problems in algebraic constructions. This is the reason why we use th…
Congruences on posets, relatively pseudocomplemented and Boolean posets
Ivan Chajda, Helmut Länger
The aim of the present paper is to extend the concept of a congruence from lattices to posets. We use an approach different from that used by the first author and V. Snášel. By usi…
Derived operators on skew orthomodular and strong skew orthomodular posets
Ivan Chajda, Helmut Länger
It is well-known that in the logic of quantum mechanics disjunctions and conjunctions can be represented by joins and meets, respectively, in an orthomodular lattice provided their…
The variety of complemented lattices where the Sasaki operations form an adjoint pair
Václav Cenker, Ivan Chajda, Helmut Länger
The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sa…
Adjointness of generalized Sasaki operations in posets
Ivan Chajda, Helmut Länger
The Sasaki projection and its dual were introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. In a previous paper the authors sh…
Quasimodules over bounded lattices
Ivan Chajda, Helmut Länger
We define a quasimodule Q over a bounded lattice L in an analogous way as a module over a semiring is defined. The essential difference is that L need not be distributive. Also for…
The operator of relative complementation
Ivan Chajda, Helmut Länger
By the operator of relative complementation is meant a mapping assigning to every element x of an interval [a,b] of a lattice L the set x^{ab} of all relative complements of x in […
Algebras and varieties where Sasaki operations form an adjoint pair
Ivan Chajda, Helmut Länger
The so-called Sasaki projection was introduced by U. Sasaki on the lattice L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H) onto a certain sublattice of…
Tense logics based on posets
Ivan Chajda, Helmut Länger, Antonio Ledda +2
Not all logical systems can be captured using algebras. We see this in classical logic (formalized by Boolean algebras) and many-valued logics (like Lukasiewicz logic with MV-algeb…
Operators on complemented lattices
Ivan Chajda, Helmut Länger
The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several compl…
Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets
Ivan Chajda, Miroslav Kolařík, Helmut Länger
On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if…
A characterization of permutability of 2-uniform tolerances on posets
Ivan Chajda, Helmut Länger
Tolerance relations were investigated by several authors in various algebraic structures, see e.g. the monograph by I. Chajda. Recently G. Czédli studied so-called 2-uniform tolera…
Implications in pseudocomplemented and Stone lattices
Ivan Chajda, Helmut Länger
Two kinds of the connective implication are introduced as term operations of a pseudocomplemented lattice. It is shown that they share a lot of properties with the intuitionistic i…