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 ¬x={¬α|α∈x}, (truncated) Minkowski sum x⊕y={α⊕β|α∈x,β∈y}, pointwise Łukasiewicz conjunction x⊙y=¬(¬x⊕¬y), the operators Δx=[min⁡x,min⁡x], ∇x=[max⁡x,max⁡x], and distinguished constants 0=[0,0],1=[1,1],i=A. We list a few equations satisfied by the algebra I(A)=(I(A),0,1,i,¬,Δ,∇,⊕,⊙), call IMV-algebra every model of these equations, and show that, conversely, every IMV-algebra is isomorphic to the IMV-algebra 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 I([0,1])-valuations. For any class Q of partially ordered algebras with operations that are monotone or antimonotone in each variable, we consider the generalization IQ of the MV-algebraic functor I, and give necessary and sufficient conditions for IQ to be a categorical equivalence. These conditions are satisfied, e.g., by all subquasivarieties of residuated lattices.