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How to produce S-tense operators on lattice effect algebras. (English) Zbl 1319.81013
The paper continues the study of tense operators on MV-algebras, published in [M. Botur and J. Paseka, “On tense MV-algebras”, Fuzzy Sets Syst. (2014); {doi:10.1016/j.fss.2014.06.006}] and [J. Paseka, Fuzzy Sets Syst. 232, 62–73 (2013; Zbl 1314.06016)] Here the results are extended to effect algebras.
Effect algebras offer a very general representation of quantum events, using a partial operation $$+$$ (disjunction of mutually excluding events) and orthosupplement (generalized negation) $$'$$. In lattice effect algebras, we may use total operation $$x\oplus y=x+(y\land x')$$ and its dual, $$x\odot y=(x'\oplus y')'$$.
Tense operators admit to introduce time and dynamics and express logical statements like “it is always going to be the case that ...” and “it has always been the case that ...”. It is shown how tense operators can be constructed for a given time frame. The authors study also the inverse task, called a representation theorem: for given tense operators on an effect algebra, to find the respective time frame. The authors distinguish E-tense operators, which preserve $$\oplus,\odot$$ just as E-states, and S-tense operators $$G$$, which, moreover, are interior operators, i.e., $$G(x)\leq x$$, $$G(G(x))=x$$.
A state is a finitely additive probability measure. If a state $$s$$ on a lattice effect algebra satisfies $$s(x\oplus x)=s(x)\oplus s(x)$$, resp. $$s(x)=1=s(y) \implies s(x\land y)=1$$ , it is called an E-state, resp. a Jauch-Piron state. The E-states are extremal states, thus they do not form a convex set.
Further, E-semi states are studied; these are only subadditive and Jauch-Piron E-semi states are infima of sets of E-states.
If an S-tense effect algebra $$E$$ has an order reflecting (=order determining) set of E-states and all E-states are Jauch-Piron states, then $$E$$ can be embedded into a tense MV-algebra. A representation theorem for such lattice effect algebras with E-tense operators is proved and demonstrated on an example.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06C15 Complemented lattices, orthocomplemented lattices and posets 03G25 Other algebras related to logic 06A11 Algebraic aspects of posets
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