Gejza Jenča

Slovak University of Technology in Bratislava · Department of Mathematics

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped...

We prove that there is a monadic adjunction between the category of bounded posets with involution and the category of orthomodular posets.

We prove that there is a monadic adjunction between the category of bounded posets with involution and the category of orthomodular posets.

We prove that the notion of a derived voltage graph comes from an adjunction between the category of voltage graphs and a category of group labeled graphs.

We prove that there is a monadic adjunction between the category of bounded posets and the category of pseudo effect algebras.

We prove that there is a monadic adjunction between the category of bounded posets and the category of pseudo effect algebras.

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an ef...

The category Rel is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, Rel is a monoidal category. Moreover, Rel is a locally posetal 2-category, since every homset Rel(A,B) is a poset with respect to inclusion. We examine the 2-category of monoids RelMon in this category. The morphism we use are lax...

We introduce two monads on the category of graphs and prove that their Eilenberg-Moore categories are isomorphic to the category of perfect matchings and the category of partial Steiner triple systems, respectively. As a simple application of these results, we describe the product in the categories of perfect matchings and partial Steiner triple sy...

There is a forgetful functor from the category of generalized effect algebras to the category of effect algebras. We prove that this functor is a right adjoint and that the corresponding left adjoint is the well-known unitization construction by Hedlíková and Pulmannová. Moreover, this adjunction is monadic.

The Kalmbach monad is the monad that arises from the free-forgetful adjunction between bounded posets and orthomodular posets. We prove that the category of effect algebras is isomorphic to the Eilenberg-Moore category for the Kalmbach monad.

We prove that for every lattice effect algebra, the system of all congruences generated by the prime ideals of the compatibility center separates the elements. This is a common generalization of Chang’s representation theorem from 1959 and a result of Graves and Selesnick (Colloq Math 27:21–30, 1973).

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension...

In the present paper, we introduce a proper superclass of homogeneous effect algebras. We call this superclass as 0-homogeneous effect algebras. We prove that in every 0-homogeneous effect algebra, the set of all sharp elements forms a subalgebra. Every chain-complete 0-homogeneous effect algebra is homogeneous. KeywordsEffect algebra-Homogeneous...

We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose that we are given an interval effect algebra E, a coexistent subset S of E, a witness mapping β for S, and an element t ∈ E ∖ S. We study the question whether there is a witness mapping β t for S ∪ {t} such that β t is an extension of...

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x ↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x ↓,x] is a subset of B. For every meager element (that means, an element x with x ↓ = 0), the...

Motivated by the notion of coexistence of effect-valued observables, we give a characterization of coexistent subsets of interval effect algebras.

We give a characterization of subsets of effect algebras, that can be embedded into a range of an observable. To give this characterization, we introduce a new notion of {\em compatibility support mappings.}

An MV-pair is a pair (B,G), where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain condition. Recently it was proved by one of the authors that for an MV-pair (B,G), ∼G is an effect-algebraic congruence and B/∼G is an MV-algebra. Moreover, every MV-algebra M can be represented by an MV-pair in this way....

We prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).

An MV-pair is a pair (B,G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ~ G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B,G), the effect algebra B/ ~ G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an...

We prove that every MV-effect algebra M is, as an effect algebra, a homomorphic image of its R-generated Boolean algebra. We characterize central elements of M in terms of the constructed homomorphism.

We prove that for every finite homogeneous effect algebra E there exists a finite orthoalgebra O(E) and a surjective full morphism φE:O(E)→E. If E is lattice ordered, then O(E) is an orthomodular lattice. Moreover, φE preserves blocks in both directions: the (pre)image of a block is always a block.

We prove that for every orthocomplete effect algebra E the center of E forms a complete Boolean algebra. As a consequence, every orthocomplete atomic effect algebra is a direct product of irreducible ones.

We prove that if E 1 and E 2 are σ-complete effect algebras such that E 1 is a factor of E 2 and E 2 is a factor of E 1, then E 1 and E 2 are isomorphic.

Partial abelian monoids (PAMs) are structures (), where is a partially defined binary operation with domain , which is commutative and associative in a restricted sense, and 0 is the neutral element. PAMs with the Riesz decomposition properties and binary relations with special properties on PAMs are studied. Relations with abelian groups, dimensio...

 We show that a quotient of a lattice ordered effect algebra L with respect to a Riesz ideal I is linearly ordered if and only if I is a prime ideal, and the quotient is an MV-algebra if and only if I is an intersection of prime ideals. A generalization of the commutators in OMLs is defined in the frame of lattice ordered effect algebras, such that...

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, called homogeneous effect algebras. This class includes orthoal...

In the present paper, we deal with a class of R 1-ideals of cancellative positive partial abelian monoids (CPAMs). We prove that, for I being an R 1-ideal of a CPAM P, P/I is a CPAM. The lattice of congruence relations associated with R 1-ideals is a sublattice of the lattice of all equivalence relations. Finally, we prove that an intersection o...

In this paper, we introduce subcentral ideals in the class of cancellative positive partial abelian monoids (CPAMs). Every complementary pair of subcentral ideals in a CPAM ℘ corresponds to a subdirect decomposition of ℘. If this decomposition is direct, the corresponding ideals are called central. Subcentral ideals are characterized as central ele...

In this paper, we introduce subcentral ideals in the class of cancellative positivepartial abelian monoids (CPAMs). Every complementary pair of subcentral idealsin a CPAM P corresponds to a subdirect decomposition of P. If this decompositionis direct, the corresponding ideals are called central. Subcentral ideals arecharacterized as central element...