Bipolar equations on complete distributive symmetric residuated lattices: The case of a join-irreducible right-hand side

Abstract

Bipolar max-⁎ equations, with ⁎ a triangular norm, have recently become a popular research topic embedded in the broad field of fuzzy relational equations. In this paper, we lift the work from the restrictive setting of the real unit interval — obfuscating the underlying lattice-theoretical essence — to the general setting of complete distributive symmetric residuated lattices, allowing to build upon the vast body of knowledge on unipolar sup-⁎ equations on complete distributive residuated lattices. We determine the full solution set, with particular emphasis on the extremal solutions, of a bipolar sup-⁎ equation in case the right-hand side is a join-irreducible element. The results are illustrated by means of ample examples.