PROPOSITIONAL LOGIC, FRAMES, AND FUZZY ALGEBRA

Abstract

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).