arXiv subjects
Publications and source records attributed to Anatolij Dvurečenskij.
Using ideas of Olson \cite{Ols} who showed that the system of effect operators of a Hilbert space can be ordered by the so-called spectral order such that the system of effect operators is a complete lattice. Using his ideas, we introduce a partial order, called the Olson order, on the set of bounded observables of a complete lattice effect algebra. We show that the set of bounded observables is a Dedekind complete lattice.0
We establish conditions when a certain type of the Riesz Decomposition Property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $\ell$-group if and only if the first one is linearly ordered and the second one is an $\ell$-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDP's. RDP's of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.
A lexicographic pseudo MV-algebra is an algebra that is isomorphic to an interval in the lexicographic product of a linear unital group with an arbitrary $\ell$-group. We present conditions when a pseudo MV-algebra is lexicographic. We show that a key condition is the existence of a lexicographic ideal, or equivalently, a case when the algebra can be split into comparable slices indexed by elements of the interval $[0,u]$ of some unital linearly ordered group $(H,u)$. Finally, we show that fixing $(H,u)$, the category of $(H,u)$-lexicographic pseudo MV-algebras is categorically equivalent to the category of $\ell$-groups.
Pseudo $MV$-algebras are a non-commutative generalization of $MV$-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo $MV$-algebras. We use the concepts of projectable pseudo $MV$-algebras and large pseudo $MV$-subalgebras to introduce orthocomplete pseudo $MV$-algebras. Then we apply a generalization of the Mundici's functor to an orthocompletion of an representable $\ell$-group to prove that each representable pseudo $MV$-algebra has an orthocompletion. In particular, our results are valid also for $MV$-algebras.
Kite pseudo effect algebras were recently introduced as a class of interesting examples of pseudo effect algebras using a po-group, an index set and two bijections on the index set. We represent kite pseudo effect algebras with a special kind of the Riesz decomposition property as an interval in a lexicographic extension of the po-group which solves an open problem on representation of kites. In addition, we introduce kite $n$-perfect pseudo effect algebras and we characterize subdirectly irreducible algebras which are building stones of the theory.
We study the class of pseudo BL-algebras whose every maximal filter is normal. We present an equational base for this class and we extend these results for the class of basic pseudo hoops with fixed strong unit.
The paper deals with special types of $L$-ordered set, $L$-fuzzy complete lattices, and fuzzy directed complete posets (fuzzy $dcpo$s). First, a theorem for constructing monotone maps is proved, a characterization for monotone maps on an $L$-fuzzy complete lattice is obtained, and it is proved that if $f$ is a monotone map on an $L$-fuzzy complete lattice $(P;e)$, then $\sqcap S_f$ is the least fixpoint of $f$. A relation between $L$-fuzzy complete lattices and fixpoints is found and fuzzy versions of monotonicity, rolling, fusion and exchange rules on $L$-complete lattices are stated. Finally, we investigate $Hom(P,P)$, where $(P;e)$ is a fuzzy $dcpo$, and we show that $Hom(P,P)$ is a fuzzy $dcpo$, the map $γ\mapsto \bigwedge_{x\in P}e(x,γ(x))$ is a fuzzy directed subset of $Hom(P,P)$, and we investigate its join.
In the paper we investigate a class of effect algebras which can be represented in the form of the lexicographic product $Γ(H\lex G,(u,0))$, where $(H,u)$ is an Abelian unital po-group and $G$ is an Abelian directed po-group. We study algebraic conditions when an effect algebra is of this form. Fixing a unital po-group $(H,u)$, the category of strong $(H,u)$-perfect effect algebra is introduced and it is shown that it is categorically equivalent to the category of directed po-group with interpolation. We show some representation theorems including a subdirect product representation by antilattice lexicographic effect algebras.
We study measures, finitely additive measures, regular measures, and $σ$-additive measures that can attain even infinite values on the quantum logic of a Hilbert space. We show when particular classes of non-negative measures can be studied in the frame of generalized effect algebras.
We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital $\ell$-group and an $\ell$-group that is not necessary Abelian. We introduce $(H,u)$-perfect pseudo MV-algebras and strong $(H,u)$-perfect pseudo MV-algebras, the latter ones will have a representation by a lexicographic product. Fixing a unital $\ell$-group $(H,u)$, the category of strong $(H,u)$-perfect pseudo MV-algebras is categorically equivalent to the category of $\ell$-groups.
Recently Jenei introduced a new structure called equality algebras which is inspired by ideas of BCK-algebras with meet. These algebras were generalized by Jenei and Kóródi to pseudo equality algebras which are aimed to find a connection with pseudo BCK-algebras with meet. We show that every pseudo equality algebra is an equality algebra. Therefore, we define a new type of pseudo equality algebras which more precisely reflects the relation to pseudo BCK-algebras with meet in the sense of Kabziński and Wroński. We describe congruences via normal closed deductive systems, and we show that the variety of pseudo equality algebras is subtractive, congruence distributive and congruence permutable.
We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected not necessarily with partially ordered groups, but rather with generalized pseudo effect algebras where the greatest element is not guaranteed. Starting even with a commutative generalized pseudo effect algebra, we can obtain a non-commutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. We find conditions when kite pseudo effect algebras have the least non-trivial normal ideal.
We present a new construction of a class pseudo BL-algebras, called kite pseudo BL-algebras. We start with a basic pseudo hoop $A$. Using two injective mappings from one set, $J$, into the second one, $I$, and with an identical copy $\overline A$ with the reverse order we construct a pseudo BL-algebra where the lower part is of the form $(\overline A)^J$ and the upper one is $A^I$. Starting with a basic commutative hoop we can obtain even a non-commutative pseudo BL-algebra or a pseudo MV-algebra, or an algebra with non-commuting negations. We describe the construction, subdirect irreducible kite pseudo BL-algebras and their classification.
We introduce a class of monotone $σ$-complete effect algebras, called representable, which are $σ$-homomorphic images of a class of monotone $σ$-complete effect algebras of functions taking values in the interval $[0,1]$ and with pointwise defined effect algebra operations. We exhibit different types of compatibilities and show their connection to representability. Finally, we study observables and show situations when information of an observable on all intervals of the form $(-\infty,t)$ gives full information about the observable.
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group $G$, a set $I$ and with two bijections $λ,ρ:I \to I.$ Using a clever construction on the ordinal sum of $(G^+)^I$ and $(G^-)^I,$ we can define a pseudo effect algebra which can be non-commutative even if $G$ is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected with partially ordered groups not necessarily with strong unit. In such a case, starting even with an Abelian po-group, we can obtain a noncommutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. Kites are so-called perfect pseudo effect algebras, and we define conditions when kite pseudo effect algebras have the least non-trivial normal ideal.
We study $\mathbb H$-perfect pseudo MV-algebras, that is, algebras which can be split into a system of ordered slices indexed by the elements of an subgroup $\mathbb H$ of the group of the real numbers. We show when they can be represented as a lexicographic product of $\mathbb H$ with some $\ell$-group. In addition, we show also a categorical equivalence of this category with the category of $\ell$-groups.
An observable on a quantum structure is any $σ$-homomorphism of quantum structures from the Borel $σ$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $σ$-MV-algebras, $σ$-effect algebras, Boolean $σ$-algebras, monotone $σ$-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.