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Tense operators on MV-algebras and Łukasiewicz-Moisil algebras. (English) Zbl 1136.03045
Let \((A,\vee ,\wedge,{}',0,1)\) be a Boolean algebra and two maps \( G,H:A\rightarrow A\) be given. The authors define \(F,P:A\rightarrow A\) by \(F(x)=(G(x'))'\) and \(P(x)=(H(x'))',\) for any \(x\in A.\) \((A,G,H)\) is a tense Boolean algebra if the following hold: (i) \(G(1)=H(1)=1;\) (ii) \(G(x\wedge y)=\) \(G(x)\wedge G(y)\) and \(H(x\wedge y)=\) \(H(x)\wedge H(y),\) for any \(x,y\in A;\) (iii) \(x\leq GP(x)\) and \(x\leq HF(x)\) for any \(x\in A\) (tense Boolean algebras are algebraic structures corresponding to the propositional tense logic).
If \((A,\vee ,\wedge ,{}',\varphi _{1},\dots,\varphi _{n-1},0,1)\) is an \(\text{LM}_{n}\) algebra then \((A,G,H)\) is a tense \(\text{LM}_{n}\) algebra (where \(G\) and \(H\) are defined as above) if (i), (ii), (iii) from above are verified and (iv) \(G\varphi _{i}=\varphi _{i}G,H\varphi _{i}=\varphi _{i}H\), for any \( i=1,2,\dots,n-1.\)
In this paper a representation theorem for tense \(\text{LM}_{n}\) algebras is proved and the polynomial equivalence between tense \(\text{LM}_{3}\) algebras (resp. tense \(\text{LM}_{4}\) algebras) and tense \(\text{MV}_{3}\) algebras (resp. tense \(\text{MV}_{4}\) algebras) is established. The authors study the pairs of dually-conjugated operations on MV algebras and use their properties in order to investigate how the axioms of tense operators are preserved by the Dedekind-MacNeille completion of an Archimedean MV algebra. A tense many-valued propositional calculus is developed and a completeness theorem is proved.

MSC:
03G20 Logical aspects of Łukasiewicz and Post algebras
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06D35 MV-algebras
03B45 Modal logic (including the logic of norms)
03B50 Many-valued logic
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