SOME EQUIVALENCE NUMBERS AND APPLICATIONS OF FUZZY AUTOMATION SEMI GROUPS

Résumé

This paper we give the proof of theorem of , Assuming that X
be a finite set with n elements, and let S(n,r) (1 ≤ r ≤ n) be the number of
equivalences ρ on X such that modX/r = r, to showing the equivalence of the
relation S(n,1) = S(n,n) = 1,S(n,r) = S(n−1,r −1)+rS(n−1,r) = (2 ≤
r ≤n−1) and use the information to calculate S(n,r) for (1 ≤ r ≤ n ≤ 6).
The numbers S(n,r) are the stirling number of the second kind [3]. S(n,r) is
a symmetric semi-group or symmetric inverse semi-group depending upon the
context. n is number of elements in base set and r is rank of elements. The
study of fuzzy sub semi-groups and fuzzy ideals within algebraic structures
like S(n,r). Fuzzy sub-sets of S(n,r) treating it as set of semi-groups.