Equational bases for some varieties of orthomodular lattices related to states

Mayet, R.

doi:10.1007/bf01237719

Cited by 20 publications

(17 citation statements)

References 3 publications

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“…In Mayet (1986), the result of Godowski was generalized into a general method, allowing to obtain by an effective procedure, for each OML L whithout a strong set of real-valued states, an equation holding in any OML with a strong set of realvalued states, and failing in L. As any HL admits a strong set of real-valued states, the equations obtained in this way all belong to E. In Mayet (1986) were given some examples for illustrating the method, but the corresponding equations where shown by Megill and Pavicić (2000) to be consequences of Godowski's equations. These authors have even expressed some doubts about the existence of equations obtained by this method that are not consequences of those of Godowski, but they report that, since then, they have obtained such equations (unpublished).…”

Section: Orthoarguesian Equations and Equations Related To Statesmentioning

confidence: 96%

“…But it was known for a long time that there are equations holding in otholattices of the form C(H) but not in general OMLs (see Godowski, 1981;Godowski and Greechie, 1984;Mayet, 1985Mayet, , 1986. More recently (Megill and Pavicić, 2000), some more results have been published, motivated by the possible repercussions to quantum computing problems.…”

Section: Introductionmentioning

confidence: 95%

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Equations Holding in Hilbert Lattices

Mayet

2006

Int J Theor Phys

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We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product.

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Section: Orthoarguesian Equations and Equations Related To Statesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 95%

Equations Holding in Hilbert Lattices

Mayet

2006

Int J Theor Phys

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“…In the rest of this section we shall be proving that the class SRODL is a variety. It should be noted that the central strategic line of the investigation of setrepresentable OMLs as used in [11,19,20,21] was instrumental in places. However, the presence of the operation △ required to invent some new techniques.…”

Section: Set-representable Odlsmentioning

confidence: 99%

Orthocomplemented lattices with a symmetric difference

Matoušek

2009

Algebra univers.

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Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices (ODL). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that "live" on set-representable OMLs.

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“…Let us remark in the conclusion of this note that in analogy with [13] there is a general way of describing all quasivarieties L C D in terms of implicative equalities. The methods to be used are essentially model theoretic and cover also cases of many other (more general and less quantum physic motivated) quasivarieties.…”

mentioning

confidence: 99%