Lattice Structures in Abstract Algebra Basic Properties and Operations on Fuzzy Sets

Abstract

Classical mathematical logic and crisp set theory are binary structures with Boolean algebra, and these structures tend to be insufficient in representing the uncertainty and ambiguity of the real world that are continuous. This study paper is a rigorous attempt to mathematically synthesize the lattice structures of the abstract algebra and the generalized theory of fuzzy sets to present a sound structural context to multi-valued logic. Using the order-theoretic properties and the algebraic properties of partially ordered sets, bound lattices, and distributive lattices, we derive the basic structure that we need in order to accommodate continuous membership functions. The paper provides a systematic description of the standard fuzzy operations of Zadeh, showing their conformity to the fundamental lattice axioms (e.g. commutativity, associativity, distributivity), and mathematically showing their inevitable failure to satisfy the strict complementarity of Boolean algebra.

Its fundamental mathematical synthesis is gained by considering L-fuzzy sets of Goguen, and how it is possible, by the mapping of membership values in the real-number unit interval to an arbitrary completely distributive lattice L, to rigorously model incomparable and multidimensional degrees of truth. We show with the help of the Resolution Principle that complex, multi-valued fuzzy sublattices may be totally decomposed into nested families of discrete, classical sublattices using generalized α-cuts. Lastly, the paper illustrates the extensive generality of these abstract algebraic theorems in computer science today, namely pointing out their applicability in enterprise artificial intelligence, fuzzy relational databases, and routing in distributed networks.