Interval MV-algebras and generalizations

Abstract

For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise \L ukasiewicz 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 \L ukasiewicz 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 \L ukasiewicz 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.