Compatible operations on commutative residuated lattices - Castiglioni, J. L.; Menni, Matías; Sagastume, Marta (2008)
Let L be a commutative residuated lattice and let f : Lk → 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 commu- tative 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 ∈ L - P (x, y) ≤ y} is compatible when defined. In particular, Pn (x, y) = y n → x, for natural number n, defines a family, Sn , 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 .
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Journal of Applied Non-Classical Logics, 18(4), pp. 413-425