Decompositions Of Elements In Complete Lattices And Their Applications To Resolutions Of Fuzzy Relational Equations

This paper deals with the decompositions of elements in completelattices and their applications in description of solution sets of fuzzy relationalequations. Firstly, a concept of principal divisor lattice is introduced and thestructures of complete principal divisor lattices are described. It is shownthat complete lower continuous principal divisor lattices have irredundantjoin irreducible decompositions, and elements have irredundant continuousjoin irreducible decompositions if and only if they are superdivisor elementsin complete lower continuous, modular and principal divisor lattices. Somenecessary and su?cient conditions that complete principal divisor distributivelattices have irredundant join irreducible decompositions are given. Undera special condition, it is proven that a complete distributive lattice hasirredundant join irreducible decompositions if and only if it is a lowercontinuous principal divisor lattice. Furthermore, some necessary and su?cientconditions that complete lower continuous, principal divisor lattices haveunique (resp. replaceable) irredundant join irreducible decompositions areformulated. In the following, some relationships among the elements of latticesare investigated, a concept of irredundant minimal join decomposition isdefined and some su?cient conditions that complete lattices have irredundantminimal join decompositions are established. Some necessary and su?cientconditions for the existence of irredundant minimal join decompositionsof elements are shown. The relationships between irredundant minimaljoin decompositions and irredundant join irreducible decompositions areconsidered. In particular, in complete principal divisor, modular lattices,a condition that irredundant minimal join decompositions and irredundantjoin irreducible decompositions of an element are equivalent is obtained. Incomplete strongly coatomic, Browerian lattices, the solution sets of fuzzyrelational equations are described, and a necessary and su?cient conditionthat fuzzy relational equations have minimal solutions is presented. Fromthe mapping point of view, a partition of solution sets of fuzzy relationalequations is given, and some properties of minimal solutions are shown.Further, some necessary and su?cient conditions for the solution sets ofdi?erent fuzzy relational equations to be same are proposed. In the end, fromthe algebraic structure point of view, the algebraic properties of solution sets offuzzy relational equations on complete Brouwerian lattices are studied. Somenecessary and su?cient conditions that the intersection of a solution to anysolution is a solution of fuzzy relational equations are derived, and some nec-essary and su?cient conditions for the solution sets to be a lattice are obtained.