Conditions under which the solution sets of fuzzy relational equations over complete Brouwerian lattices form lattices

It is well known that the solution set of a fuzzy relational equation with sup–inf composition is a join semilattice, in general, not a meet semilattice. This paper investigates the conditions under which the solution sets of fuzzy relational equations with sup-inf composition over complete Brouwerian lattices form lattices. We first give some properties of the decompositions of elements in complete lattices, then present a necessary and sufficient condition that the meet of a solution with any other solution is again a solution. Finally, we show some necessary and sufficient conditions that the solution sets are lattices.