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Interval MV-algebras and generalizations. (English) Zbl 1326.06011
In this paper, the set of all intervals of an MV-algebra $$A$$ is equipped with the following operations and constants: $$\neg I=\{\neg x\mid x\in I\}$$; $$I\oplus J=\{x\oplus y\mid x\in I,\;y\in J\}$$; $$\Delta(I)=[\min I,\min I]$$; $$\nabla I=[\max I,\max I]$$; $$0=[0,0]$$; $$1=[1,1]$$; and $$i=A$$. The resulting algebraic structure is called $$I(A)$$, the interval algebra of $$A$$; moreover the interval algebra construction is functorial. The models of certain finitely many equational properties of $$I(A)$$ are called IMV-algebras.
It is shown that every IMV-algebra is isomorphic to $$I(B)$$ for some MV-algebra $$B$$. The category of IMV-algebras is shown to be equivalent to the category of MV-algebras, and its free objects are characterized. Then Łukasiewicz interval logic is defined as the deductive system whose inference rules are semantic consequence relations between IMV-terms. The tautology and consequence problems for this logic are coNP-complete.
Finally a vast generalization of the interval algebra construction is performed: instead of MV-algebras, a large class of quasivarieties of partially ordered algebras is considered, where the operations are monotone or antimonotone in each variable. It results that the interval algebra functor is an equivalence for many quasivarieties, and necessary and sufficient conditions are given for this to happen.

##### MSC:
 06D35 MV-algebras 03G25 Other algebras related to logic
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