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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