Fuzzy binary relations

Morsi, N. N. (1994) Hyperspace fuzzy binary relations Fuzzy Sets and Systems. 67(2) 221—37. [The author associates with each implication operator in (0, l)-valued logic, under certain conditions, an algorithm for extending a fuzzy or ordinary binary relation psi from X to Y, to a fuzzy binary relation from 1(20 to 1(f) said to be a fuzzy hyperspace extension of psi.]... 

Concepts that are important for general n-dimensional fuzzy relations include projections to lower-dimensional spaces, cylindric extensions of projections, and cylindric closures. These concepts are simple generalizations of their classical counterparts, and it is not essential to cover them here. It is more important to introduce some key concepts regarding fuzzy binary relations, which have a broad applicability. 

Membership functions of any fuzzy binary relation on X Y have the form... 

Consider fuzzy binary relations P, Q, and R, defined on sets XxY, YxZ, and X X Z, respectively, for which... 

Because they map onto [0,1], similarity relations are fuzzy relations [50], which differ from classical relations that map pairs of elements onto the set of binary values 0,1. Similarity relations satisfy two mathematical properties, namely, they are reflexive, S(i,j)=0 if m =m., and generally are symmetric, S(iJ)=S(j,i) for l ij n, but they are generally intransitive. Asymmetric similarities, which will be discussed in Section 15.5.1, have been employed in MSA, but the number of applications is relatively small to date. 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

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