Many-valued quantum algebras

Chajda, Ivan; Halaš, Radomír; Kühr, Jan

doi:10.1007/s00012-008-2086-9

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In our previous paper [1] we introduced the concept of a basic algebra, this being an algebra (A, ⊕, ¬, 0) of type (2, 1, 0) with the property that the underlying poset (A, ≤) defined by x ≤ y if and only if ¬x ⊕ y = ¬0 is a bounded lattice and, for each a ∈ A, the mapping x → ¬x ⊕ a is an antitone involution on the principal filter [a) = {x ∈ A | a ≤ x}. The name 'basic algebra' is used because these algebras capture common features of many known structures such as Boolean algebras, orthomodular lattices, MV-algebras or lattice effect algebras.

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mentioning

confidence: 94%

Every effect algebra can be made into a total algebra

Chajda

¹

,

Halaš

²

,

Kühr

³

2009

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In our previous paper [1] we introduced the concept of a basic algebra, this being an algebra (A, ⊕, ¬, 0) of type (2, 1, 0) with the property that the underlying poset (A, ≤) defined by x ≤ y if and only if ¬x ⊕ y = ¬0 is a bounded lattice and, for each a ∈ A, the mapping x → ¬x ⊕ a is an antitone involution on the principal filter [a) = {x ∈ A | a ≤ x}. The name 'basic algebra' is used because these algebras capture common features of many known structures such as Boolean algebras, orthomodular lattices, MV-algebras or lattice effect algebras. In [1] we paid special attention to lattice effect algebras, which were originally defined as partial algebras (E, +, 0, 1), but the presence of the join operation allows one to replace partial + by total ⊕. The intent of the present paper is to establish similar results for general effect algebras in the context of commutative directoids; we shall prove that every effect algebra (E, +, 0, 1) can be made into a total algebra (E, ⊕, ¬, 0) in such a way that two elements are compatible in (E, +, 0, 1) if and only if they commute in (E, ⊕, ¬, 0).

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“…We recall that an MV-algebra is an algebra M = (M, +, ¬, 0) of type (2, 1, 0) satisfying identities (M1) -(M6): [7] or Example 4.4 in [2]). …”

Section: ) Is a Lattice Orthoalgebra If And Only If Its Correspondingmentioning

confidence: 99%

“…A representation of lattice effect algebras by means of so-called basic algebras was derived in [2].…”

mentioning

confidence: 99%

A Representation of Lattice Effect Algebras by Means of Right Near Semirings with Involution

Chajda

¹

,

Länger

²

2016

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Since every lattice effect algebra decomposes into blocks which are MV-algebras and since every MV-algebra can be represented by a certain semiring with an antitone involution as shown by Belluce, Di Nola and Ferraioli, the natural question arises if a lattice effect algebra can also be represented by means of a semiring-like structure. This question is answered in the present paper by establishing a one-to-one correspondence between lattice effect algebras and certain right near semirings with an antitone involution.Keywords Effect algebra · Lattice effect algebra · Right near semiring · Antitone involution · Effect near semiring Effect algebras were introduced by Foulis and Bennett [8] in order to axiomatize unsharp logics of quantum mechanics. Although the definition of an effect algebra looks elementary, these algebras have several very surprising properties. Concerning these properties the reader is referred to the monograph [7] by Dvurečenskij and Pulmannová. In particular, every effect algebra induces a natural partial order relation and thus can be considered as a bounded poset. If this poset is a lattice, the effect algebra is called a lattice effect algebra.

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“…/_y and :x D x ? , and constitute as well as provide an axiomatization of the logic of quantum mechanics along with MV-algebras [14], which get an axiomatization of many-valued Łukasiewicz logics; see Chajda [15] and Chajda et al [16].…”

Section: Introductionmentioning

confidence: 99%