PSEUDO-COMPLEMENTATION ON ALMOST DISTRIBUTIVE FUZZY LATTICES

The general development of lattice theory started by G. Birkhoff [1]. The concept of an Almost distributive lattice (ADL) was introduced by U.M. Swamy and G.C. Rao [2] as a common abstraction of almost all the existing ring theoretic
generalizations of a Boolean algebra. The structure of pseudo complemented distributive lattice I and II given by H. Lakser [8, 9] and G. Gratzer [9]. In [4] A. Berhanu, G. Yohannes and T. Bekalu introduced Almost distributive fuzzy lattice (ADFL). In [5] K. B. Lee proved that any Pseudo-complementation on a semilattice is equationally definable. The notion of Pseudocomplementation in an almost distributive lattices was introduced by U.M. Swamy, G.C. Rao and G.N. Rao in [3] and they observe that an almost distributive lattices have more than one pseudo-complementation while it is unique in case of
distributive lattice. Pseudo-complements in semi-lattices
introduced by O. Frink [6] and also by A.F, Lopez and M.I.T.
Barrosa [7]. On the other hand, L.A. Zadeh [12] introduced Fuzzy
sets to describe vagueness mathematically in its very
abstractness and tried to solve such problems by assigning to
each possible individual in the universe of discourse a value
representing its grade of membership in the fuzzy set. In [13] N.
Ajmal and K.V. Thomas defined a Fuzzy lattice as a fuzzy algebra
and characterized fuzzy sublattices. I. Chon [14] considering the notion of fuzzy order of Zadeh, introduced a new notion of Fuzzy lattices and fuzzy partial order relations. In this paper, we introduce the concept of PseudoComplementation ∗ on an ADFL and prove that it is equationally definable in ADFL. We characterized properties of PseudoComplementation on Almost distributive fuzzy lattice (PCADFL)
and we give some preliminary results in PCADFL.