Regarding Powers of Real Gödel-Kleene Lattices
We present the known notion of Gödel-Kleene lattice (LK-lattice) and its basic properties in order to show its potential interest in fuzzy set theory for membership functions computation. A structure of LK-lattice with special operators is introduced on any cartesian power of the real LK-chain. On this basis, an algebraic modal logic of multiple criteria real-valued optimization is proposed. Then a connection with the original paper of Bellman and Zadeh on decision making in fuzzy environment will be established. The main aim of this paper is to present an algebraic theory for the case of two-criteria optimization, in order to reflect some basic properties of the cartesian square of the real LK-chain together with the special operators previously introduced. A starting point in this direction, is considered to be the notion of bisymmetric Gödel-Kleene lattice (S2LK-lattice) defined by a 2-cyclic LK-lattice. We show that on the LK-lattice reduct of any S2LK-lattice, there exist a universal quantifier and an existential quantifier, which define a structure of S5-type modal LK-lattice. Then, with respect to the main aim, we introduce a new algebraic structure, called modal bisymmetric Gödel-Kleene algebra (μS2LK-algebra), which is defined by an S2LK-lattice together with a pair of special modal endomorphisms. One obtains that the cartesian square of any LK-lattice can be equipped with a standard structure of μS2LK-algebra. Different properties of μS2LK-algebras are presented. A generalization of the notion of intutionistic fuzzy set is derived.
Keywords: De Morgan lattice, Kleene lattice, symmetric Heyting algebra, Gödel-Kleene lattice, multiple criteria decision making, fuzzy set, involutive Brouwerian D-algebra.