Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Mathematica Slovaca

Hull mappings and dimension effect algebras

David J. Foulis, Sylvia Pulmannová

Published May 14, 2011• 12 citations
Publisher

Abstract

Abstract An effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.

DOI
10.2478/s12175-011-0025-2

Related papers

TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

David J. Foulis, Sylvia Pulmannová, Elena Vinceková

Abstract Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

Journal of the Australian Mathematical Society•Dec 1, 2010•10 citations

From Boolean algebras to effect algebras: Sharp and unsharp logic in classical and quantum systems

Atul Kumar Singh, Akhilesh Kumar Singh

This paper examines how logical structures represent sharp and unsharp propositions in classical and quantum systems. Classical sharp logic is modeled by Boolean algebras, where propositions have bivalent truth values, complements are globally defined, and distributivity holds. Quantum sharp logic is modeled by orthomodular lattices of closed Hilbert-space subspaces or projection operators; it retains definite yes–no outcomes for ideal measurements but rejects the Boolean assumptions of global truth valuation and distributivity. Classical unsharp logic uses fuzzy membership or probabilistic truth values to describe vagueness and incomplete information, whereas quantum unsharp logic is expressed through effects and effect algebras, which capture noisy, inefficient, and generalized measurements. The paper’s central question is which algebraic structure is appropriate for each combination of classical or quantum behavior and sharp or unsharp measurement. The analysis shows that the four cases form a coherent hierarchy: Boolean algebras describe deterministic classical events, orthomodular lattices describe ideal quantum propositions, fuzzy or probabilistic models describe classical partial truth, and effect algebras describe realistic quantum effects. This classification clarifies the mathematical and physical role of unsharpness and explains why effect algebras are needed beyond projection-based quantum logic.

Engineering and Applied Science Letters•Aug 12, 2026•0 citations

Sublattices and Δ-blocks of orthomodular posets

Ivan Chajda, Helmut Länger

Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

arXiv:2004.01962•Journal of Logic and Computation•Jun 25, 2020•0 citations

Effect algebras and statistical physical theories

Enrico G. Beltrametti, Slawomir Bugajski

The dichotomic physical quantities of a physical system can be naturally hosted in a mathematical structure, called effect algebra, of which orthomodular posets and Boolean algebras are particular examples. We examine how effect algebras arise inside statistical physical theories and, conversely, we study to what extent an effect algebra can be taken as a primitive structure on which a satisfactory statistical physical model equipped with a convex set of states can be constructed.

Journal of Mathematical Physics•Jun 1, 1997•42 citations

An axiomatisation of quantum logic

Ian Clark

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.

Journal of Symbolic Logic•Sep 1, 1973•13 citations

Effect algebras, witness pairs and observables

Gejza Jenča

The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a witness pair for a subset of S is an object living within E that "witnesses existence" of an observable whose range includes S. We prove that there is an adjunction between the poset of all witness pairs of E and the category of all partially inverted E-valued observables.

EPiC series in computing•Jan 23, 2018•0 citations
Abstract An effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.
Abstract Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.
This paper examines how logical structures represent sharp and unsharp propositions in classical and quantum systems. Classical sharp logic is modeled by Boolean algebras, where propositions have bivalent truth values, complements are globally defined, and distributivity holds. Quantum sharp logic is modeled by orthomodular lattices of closed Hilbert-space subspaces or projection operators; it retains definite yes–no outcomes for ideal measurements but rejects the Boolean assumptions of global truth valuation and distributivity. Classical unsharp logic uses fuzzy membership or probabilistic truth values to describe vagueness and incomplete information, whereas quantum unsharp logic is expressed through effects and effect algebras, which capture noisy, inefficient, and generalized measurements. The paper’s central question is which algebraic structure is appropriate for each combination of classical or quantum behavior and sharp or unsharp measurement. The analysis shows that the four cases form a coherent hierarchy: Boolean algebras describe deterministic classical events, orthomodular lattices describe ideal quantum propositions, fuzzy or probabilistic models describe classical partial truth, and effect algebras describe realistic quantum effects. This classification clarifies the mathematical and physical role of unsharpness and explains why effect algebras are needed beyond projection-based quantum logic.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ\mathrel \DeltaΔ and a binary operator d(x,y)d(x,y)d(x,y) that are generalizations of the binary relation C\textrm{C}C and the commutator c(x,y)c(x,y)c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ\mathrel \DeltaΔ and ↔\leftrightarrow↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y)d(x,y)d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ\DeltaΔ-blocks with respect to Boolean subalgebras and distributive subposets they include.
The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.
The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a witness pair for a subset of S is an object living within E that "witnesses existence" of an observable whose range includes S. We prove that there is an adjunction between the poset of all witness pairs of E and the category of all partially inverted E-valued observables.