Henry Martin

In this paper, we introduce and study the variety of algebras (A,∧,∨,→,◻,◊,0,1) of type (2,2,2,1,1,0,0) whose {∧,∨,→,0,1}-reduct is a weak Heyting algebra and the following identities are satisfied: (a) ◻1=1, (b) ◻(a∧b)=◻a∧◻b, (c) ◊0=0 and (d) ◊(a∨b)=◊a∨◊b. This variety, which is denoted by MWH, contains several varieties of Heyting algebras with modal operators, which are the algebraic semantics of well-known modal intuitionistic logics. The main goal of this paper is to study certain subvarieties of MWH, which are the algebraic semantics of particular subintuitionistic modal logics. We give representation theorems for these subvarieties, we prove that they are canonical and we also show that their associated deductive systems are frame complete with respect to certain class of relational frames.

This work extend to residuated lattices the results of [ 7 ]. It also provides a possible generalization to this context of frontal operators in the sense of [ 9 ]. Let L be a residuated lattice, and f : L k → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P ( x, y ) and Q ( x, y ) on a residuated lattice L which imply that the function $${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators.