Every effect algebra can be made into a total algebra

Chajda, Ivan; Halaš, Radomír; Kühr, Jan

doi:10.1007/s00012-009-0010-6

Cited by 29 publications

(24 citation statements)

References 6 publications

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“…[1], [3], [4]) is an algebra A = (A; ⊕, ¬, 0) of type (2,1,0) Having an arbitrary basic algebra A = (A; ⊕, ¬, 0), the restriction of the binary operation ⊕ to orthogonal elements yields a partial binary operation + such that (A; +, 0, 1) becomes a weak effect algebra. Unfortunately, weak effect algebras cannot be represented by means of basic algebras since every basic algebra induces a lattice order but weak effect algebras need not be lattice ordered (see [1]).…”

Section: Ivan Chajdamentioning

confidence: 99%

“…We will need the following concept of a weak basic algebra (introduced by R. Halaš and L. Plojhar [7]). Instead of the original axiomatic system from [7], we will use an equivalent one from [2] which is a bit more simple.…”

Section: Ivan Chajdamentioning

confidence: 99%

“…It was proved in [2] that (A; ) is a commutative directoid (in the sense of [8]), i.e. it is characterized by the identities…”

Section: èöóôó× ø óò 2ºmentioning

confidence: 99%

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A representation of weak effect algebras

Chajda

2012

Mathematica Slovaca

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ABSTRACT. Weak effect algebras were introduced by the author as a generalization of effect algebras and pseudoeffect algebras. It was shown that having a basic algebra, we can restrict its binary operation to orthogonal elements only and what we get is just a weak effect algebra. However, the converse construction is impossible due to the fact that the underlying poset of a basic algebra is a lattice which need not be true for weak effect algebras. Hence, we found a weaker structure than a basic algebra which can serve as a representation of a weak effect algebra. The concept of effect algebra was introduced by D. J. Foulis and M. K. Bennett [9] for the algebraic axiomatization of unsharp reasoning in quantum logics. A. Dvurečenskij and T. Vetterlein generalized this concept by dropping commutativity of the partial binary operation. Their structure is called a pseudoeffect algebra and was introduced in [6]. It was shown in [2] that the partial binary operation of an effect algebra can be extended into a total operation. The resulting structure is a so-called weak basic algebra. This weak basic algebra represents an effect algebra if and only if it satisfies a rather complex identity which in fact describes the condition saying that for orthogonal elements, the partial operation addition is associative and commutative (which need not be true for the total operation). Dropping this assumption, we can derive more general structures, so-called weak effect algebras (see [1]). Hence weak effect algebras form a generalization of effect algebras where also associativity is deleted.2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06A06, 06A12, 06D35, 03G10, 03G12. K e y w o r d s: weak effect algebra, weak basic algebra, directoid, antitone involution.

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Section: Ivan Chajdamentioning

confidence: 99%

Section: Ivan Chajdamentioning

confidence: 99%

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A representation of weak effect algebras

Chajda

2012

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“…It was shown in [8] and [9] that lattice effect algebras are induced by a much more general structures, the so-called basic algebras satisfying certain identities. For effect algebras which do not have a lattice order this is done in [9]. Our aim is to show what an effect-like algebra can be induced by a basic algebra if no additional conditions are assumed.…”

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Effect-like algebras induced by means of basic algebras

Chajda

2009

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ABSTRACT. Having an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied. . An effect algebra is a partial algebra which serves as a generalization of the set of Hilbert-space effects, i.e. self-adjoint operators on a Hilbert space (see e.g.[11] for the motivation in full details). For reader's convenience, we recall the definition of effect algebra. Ò Ø ÓÒ 1ºAn effect algebra is a partial algebra E = (E; +, 0, 1) of type (2, 0, 0) satisfying the axioms (EA1) if x + y is defined then y + x is defined and x + y = y + x; (EA2) if x + y and (x + y) + z are defined then y + z and x + (y + z) are defined and x + (y + z) = (x + y) + z;(EA3) for each a ∈ E there exists a unique b ∈ E such that a + b = 1; let us denote this b by a ; (EA4) if 1 + a is defined then a = 0.2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 06B05, 06B15, 06D35, 06D30, 03G25, 03G10, 06A12. K e y w o r d s: effect algebra, MV-algebra, basic algebra, weak effect algebra, commutative weak effect algebra.

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“…in orthomodular lattices · is associative and commutative while is not. Sometimes it can be useful to study right residuated (or even residuated) structures by the partial operations [2].…”

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Open projections do not form a right residuated lattice

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EPiC Series in Computing

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Every effect algebra can be made into a total algebra

Cited by 29 publications

References 6 publications

A representation of weak effect algebras

A representation of weak effect algebras

Effect-like algebras induced by means of basic algebras

Open projections do not form a right residuated lattice

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