Representation theory for complete L-lattices

Abstract

In the framework of L-valued (fuzzy) sets, where L is a complete lattice, we introduce complete L-lattices, based on L-structures investigated by the authors. An L-poset is a set equipped with an L-valued equality E and an L-valued transitive relation R, which is antisymmetric with respect to E. A complete L-lattice is an L-poset in which every subset has a so called pseudo-supremum and a pseudo-infimum. Several properties concerning special elements of these L-structures are investigated. Among our main results, we prove that an L-poset is a complete L-lattice if and only if particular quotient substructures with respect to the L-valued equality are classical complete lattices. As another important result obtained by using closure systems, we present a Representation theorem dealing with a general construction of L-posets and Lcomplete lattices.