Residual transfer in fuzzy algebraic structures

Abstract

This thesis attempts a combination of three important areas of mathematics, namely universal algebra, residuation theory and fuzzy set theory. A fuzzy subalgebra of a universal algebra A := (A; FA) of type F under a residuated lattice L := (L; ^; _; _; (; 0; 1), called an L-fuzzy subalgebra of A, is a map from A to L which is ^-compatible with the fundamental operations of A. This notion was introduced by V. Murali [29] in 1991, under the unit interval [0; 1] of real numbers, and generalized by B. _ Se_selja [35] in 1996, under partially ordered sets. Given a residuated lattice L and a universal algebra A of type F with a residuated lattice Sub(A) :=Sub(A); \; t; _; !; ; Sg(;); A _on the set of its subuniverses, the set Fu(A; L) of L-fuzzy subsets of A forms a residuated lattice Fu(A; L) :=Fu(A; L); ^; _; _; (; 0; 1_ that extends both L and the Boolean algebra P(A) of subsets of A. The set Fs(A; L) 􀀀 of L-fuzzy subalgebras of A forms a bounded lattice Fs(A; L) :=Fs(A; L); ^; t; _Sg(;); 1_, but not necessarily a residuated lattice, which extends both the bounded lattices of L and Sub(A). When L is a finite linearly ordered Brouwerian algebra, Fs(A; L) forms an algebraic residuated lattice Fs(A; L) :=Fs(A; L); ^; t; ; ,!; #; _Sg(;); 1_that extends both L and Sub(A). The condition on the residuated lattice L of the preceding result being rather restrictive, it is natural to look for some classes of algebras for which the latter is more general. In this thesis, two solutions to this problem are proposed, in the classes of mono-unary algebras and rings, and some of their properties are investigated.