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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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