On the weakability of fuzzy neighborhood systems on various algebraic structures,

Abstract

Let X, X′ be two locally finite, preordered sets and let R be any indecomposable commutative ring. The incidence algebra I(X,R), in a sense, represents X, because of the well-known result that if the rings I(X,R) and I(X′,R) are isomorphic, then X and X′ are isomorphic. In this paper, we consider a preordered set X that need not be locally finite but has the property that each of its equivalence classes of equivalent elements is finite. Define I*(X,R) to be the set of all those functions f:X×X→R such that f(x,y)=0, whenever x⩽̸y and the set Sf of ordered pairs (x,y) with x<y and f(x,y)≠0 is finite. For any f,g∈I*(X,R), r∈R, define f+g, fg, and rf in I*(X,R) such that (f+g)(x+y)=f(x,y)+g(x,y), fg(x,y)=∑x≤z≤yf(x,z)g(z,y), rf(x,y)=r⋅f(x,y). This makes I*(X,R) an R-algebra, called the weak incidence algebra of X over R. In the first part of the paper it is shown that indeed I*(X,R) represents X. After this all the essential one-sided ideals of I*(X,R) are determined and the maximal right (left) ring of quotients of I*(X,R) is discussed. It is shown that the results proved can give a large class of rings whose maximal right ring of quotients need not be isomorphic to its maximal left ring of quotients.