Orthomodular (Partial) Algebras and Their Representations

Burmeister, Peter; Mączyński, Maciej

doi:10.1515/dema-1994-3-415

Cited by 15 publications

(25 citation statements)

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“…In [BM94] we have given an axiomatic description of the class of all orthomodular (partially) ordered sets as an ECE-variety of what we have called orthomodular (partial) algebras. For the convenience of the reader we repeat here the definition of an orthomodular (partial) algebra as given in [BM94].…”

Section: Basic Properties Revisitedmentioning

confidence: 99%

“…For the convenience of the reader we repeat here the definition of an orthomodular (partial) algebra as given in [BM94]. DEFINITION 1.1.…”

Section: Basic Properties Revisitedmentioning

confidence: 99%

“…In [BM94] we have started to develop the theory of orthomodular (partial) algebras (briefly: OMAs), and in this paper we continue this development. Among others we want to study the connection of OMAs with quasi-rings, and to investigate the structure of the ordered sets of congruence relations of OMAs.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-rings and congruences in the theory of orthomodular algebras

Burmeister

Holzer

Mączyński

2000

Algebra Universalis

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In this paper the theory of orthomodular algebras (OMAs) is further developed. The connection with quasi-rings and the structure of the ordered sets of their congruence relations are investigated. The paper is divided into five sections: on the structure of OMAs, the extensions of orthomodular addition, orthomodular algebras and orthomodular quasi-rings, OMA congruences, and OMA-Schmidt congruences. In conclusion, all OMA-Schmidt congruence relations for very small OMAs are computed and the line diagram of the OMA-Schmidt congruence lattice for the OMA with two free generators is constructed.

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Section: Basic Properties Revisitedmentioning

confidence: 99%

“…For the convenience of the reader we repeat here the definition of an orthomodular (partial) algebra as given in [BM94]. DEFINITION 1.1.…”

Section: Basic Properties Revisitedmentioning

confidence: 99%

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Quasi-rings and congruences in the theory of orthomodular algebras

Burmeister

Holzer

Mączyński

2000

Algebra Universalis

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“…For a theory of orthoalgebras see e. g. [5], for a theory of orthomodular algebras see e. g. [2]. For orthoalgebras we also have where V denotes the supremum with respect to the partial order < on A which was defined above.…”

Section: An Application To Orthomodular Posets (Quantum Logics)mentioning

confidence: 99%

The Logic Induced by a System of Homomorphisms and Its Various Algebraic Characterizations

Dorninger¹,

Länger²,

Mączyński³

1997

Demonstratio Mathematica

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IntroductionThe well-known spectral theorem for self-adjoint operators on a Hilbert space can be formulated as follows:Let H be a complex separable Hilbert space with dim H > 2 and let L(H) denote the orthomodular lattice (shortly, OML) of all orthogonal projections from H onto closed linear subspaces of H. Let O denote the set of all self-adjoint linear operators on H and {m Q \ a (E S} the set of all pure probability measures on L(H). Then for every A 6 O there exists a unique Z(//)-valued measure (spectral measure) HA on B(R) such that for every a E S the composed mapping m a o is a probability measure on B(R). (Here and in the following B(R) denotes the Boolean a-algebra of all Borel sets of the real line.) Hence, the spectral theorem determines a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R) such that each Pa,o can be decomposed in the form p A ,a = f^a 0 Pa where /i^ is an £(#)-valued measure on B(R) and m a is a pure probability measure on L(H). The family (PA,a)Aeo,aes can be interpreted as the spectral family of probability measures on B(R) corresponding to O and S. Now, by the inverse spectral theorem we may understand the following problem:Given a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R), what conditions are to be put on O and S in order that every This paper is a result of the collaboration of the three authors within the framework of the Partnership Agreement between the University of Technology Vienna and the Warsaw University of Technology. The authors are grateful to both universities for providing financial support which has made this collaboration possible.

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“…The second level is a cancellative PAM (CPAM) and the third level is an effect algebra (or D-poset). Effect algebras [8,10,11], D-posets [5,6,15] and related structures [1,4,14,16,18] have been introduced as abstract models for studying quantum effects and unsharp quantum measurements. These structures are more general than previously considered frameworks in the quantum logic approach to the foundations of quantum theory [2,3,12,13,17].…”

Section: Introductionmentioning

confidence: 99%

Quotients of partial abelian monoids

Gudder¹,

Pulmannov�²

1997

Algebra Universalis

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A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and orthomodular lattices (OML). If P is a PAM, the concepts of a congruence on P and a quotient P// are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts.

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Orthomodular (Partial) Algebras and Their Representations

Cited by 15 publications

References 0 publications

Quasi-rings and congruences in the theory of orthomodular algebras

Quasi-rings and congruences in the theory of orthomodular algebras

The Logic Induced by a System of Homomorphisms and Its Various Algebraic Characterizations

Quotients of partial abelian monoids

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