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De Morgan algebras with tense operators. (English) Zbl 1398.06008
Summary: To every propositional logic satisfying double negation law is assigned a De Morgan poset $$\mathcal E$$. Using of axioms for an universal quantifier, we set up axioms for the so-called tense operators $$G$$ and $$H$$ on $$\mathcal E$$. The triple $$\mathcal D=(\mathcal E;G,H)$$ is called a (partial) dynamic De Morgan algebra.
We solve the following questions: first, if a time frame is given, how to construct tense operators $$G$$ and $$H$$; second, if a (strict) dynamic De Morgan algebra is given, how to find a time frame such that its tense operators $$G$$ and $$H$$ can be reached by this construction. In particular, any strict dynamic De Morgan algebra is representable in its Dedekind-MacNeille completion with respect to a suitable countable time frame.
##### MSC:
 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) 03G10 Logical aspects of lattices and related structures 03B44 Temporal logic
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