Compatible operations on commutative residuated lattices

Abstract

Let L be a commutative residuated lattice and let 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 commutative residuated lattices is locally affine complete. Then, we find conditions on a not necessarily polynomial function P(x, y) in L that imply that the function x ↦ min{y є L | P(x, y) ⪯ y} is compatible when defined. In particular, P_n(x, y) = yⁿ → x, for natural number n, defines a family, S_n, of compatible functions on some commutative residuated lattices. We show through examples that S₁> and S₂, defined respectively from P₁ and P₂, are independent as operations over this variety; i.e. neither S₁ is definable as a polynomial in the language of L enriched with S₂ nor S₂ in that enriched with S₁.

Keywords:

• commutative residuated lattices
• compatible operations
• affine completeness