Equational characterization for two-valued states in orthomodular quantum systems

Domenech, Graciela; Freytes, Héctor; Ronde, Christian de

doi:10.1016/s0034-4877(11)60027-x

Cited by 6 publications

(2 citation statements)

References 35 publications

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“…Next, we list some examples of factor varieties. Quasi-discriminator varieties include discriminator varieties as well as many nondiscriminator examples, like (1) Glivenko MTL algebras with the Boolean retraction property [12], hence in particular Gödel algebras, Product algebras [19], and the variety of MV algebras generated by Chang's algebra [11]; (2) regular Nelson residuated lattices [9]; (3) Jauch-Piron orthomodular lattices with states [15].…”

Section: Factor Varietiesmentioning

confidence: 99%

Factor varieties

2015

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The universal algebraic literature is rife with generalisations of discriminator varieties, whereby several investigators have tried to preserve in more general settings as much as possible of their structure theory. Here, we modify the definition of discriminator algebra by having the switching function project onto its third coordinate in case the ordered pair of its first two coordinates belongs to a designated relation (not necessarily the diagonal relation). We call these algebras factor algebras and the varieties they generate factor varieties. Among other things, we provide an equational description of these varieties and match equational conditions involving the factor term with properties of the associated factor relation. Factor varieties include, apart from discriminator varieties, several varieties of algebras from quantum and fuzzy logics

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Section: Factor Varietiesmentioning

confidence: 99%

Factor varieties

2015

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“…Based on the above mentioned two properties, in [5], Boolean pre-states are introduced as a general theoretical framework to study families of two-valued states on orthomodular lattices. We shall use these ideas for a general study of two-valued states extended to Baer * -semigroups.…”

Section: Basic Notionsmentioning

confidence: 99%

Two-Valued States on Baer *-Semigroups

Freytes

Domenech

Ronde

2013

Reports on Mathematical Physics

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In this paper we develop an algebraic framework that allows us to extend families of two-valued states on orthomodular lattices to Baer *semigroups. We apply this general approach to study the full class of two-valued states and the subclass of Jauch-Piron two-valued states on Baer * -semigroups. Proposition 2.1 Let L be an orthomodular lattice. Then we have: 1. Z(L) is a Boolean sublattice of L [20, Theorem 4.15].2. z ∈ Z(L) iff for each a ∈ L, a = (a ∧ z) ∨ (a ∧ ¬z) [20, Lemma 29.9].

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