Interval MV-algebras and generalizations

Abstract

For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise Łukasiewicz negation $\neg x=\{\neg \alpha \mid \alpha \in x\}$, (truncated) Minkowski sum, $x\oplus y=\{\alpha \oplus \beta \mid \alpha \in x,\beta \in y\}$, pointwise Łukasiewicz conjunction $x\odot y=\neg (\neg x\oplus \neg y)$, the operators $\Delta x=[\min x,\min x]$, $\nabla x=[\max x,\max x]$, and distinguished constants $0=[0,0],1=[1,1],\mathsf{i}=A$. We list a few equations satisfied by the algebra $\mathcal{I}(A)=(I(A),0,1,\mathsf{i},\neg ,\Delta ,\nabla ,\oplus ,\odot )$, call IMV-algebra every model ofthese equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra $\mathcal{I}(B)$ of all intervals in some MV-algebra $B$. We show that IMV-algebras are categorically equivalent to MV-algebras, and give a representation of free IMV-algebras. We construct Łukasiewicz interval logic, with its coNP-complete consequence relation, which we prove to be complete for $\mathcal{I}([0,1])$-valuations. For any class $\mathsf{Q}$ of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization ${\mathcal{I}}_{\mathsf{Q}}$ of the MV-algebraic functor $\mathcal{I}$, and give necessary and sufficient conditions for ${\mathcal{I}}_{\mathsf{Q}}$ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.

References in corpus (1)

• On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice