Interval MV-algebras and generalizations

For any MV-algebra $A$ we equip the set $I(A)$ of intervals in $A$ with pointwise \L ukasiewicz negation $\mathrm{\neg }x=\{\mathrm{\neg }\alpha \mid \alpha \in x\}$, (truncated) Minkowski sum, $x\oplus y={\alpha\oplus \beta\mid \alpha \in x,,,\beta\in y}$,pointwise\text{L}ukasiewiczconjunction$x\odot y=\neg(\neg x\oplus \neg y)$,theoperators$\Delta x=[\min x,\min x]$,$\nabla x=[\max x,\max x]$,anddistinguishedconstants$0=[0,0],,, 1=[1,1],,,, $\mathsf{i}$ = A$\mathrm{.}Welistafewequationssatisfiedbythealgebra$\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}$}$oftheMV-algebraicfunctor$\mathcal I$, 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.