A Note on the Solutions to Lattice-valued Relational Equations and Inequations

Abstract

The existence of the greatest solutions to some lattice-valued relational equations, inequations and their systems is proved in the case of a complete residuated codomain lattice. The aim is to structure the existing knowledge and to provide missing results related to equations and inequations, where there is only one unknown on every side of the (in)equation. We also prove the existence of the least solution, for some equations and inequations. By a counter-example, we prove that the minimal solution of some typical equations need not exist. We also prove some more general results in the case where we have more than one variable on one, or both sides, of the (in)equations. In some cases, when the greatest solution need not exist, we prove the existence of a maximal solution. A procedure yielding one such solution, is also offered in this paper.