hK)TE ON SEMIIAXTICE ~UENCES
Zensiro Goseki · 1976
Throughout this note S will denote a semigroup. As is well known, there is the smallest semilattice conon S in the sense of inclusion. If O ° = S is called s-ind~sable. The pattidue to 0 ° is called the greatest s-decc~S. In the greatest s-d~sition of S each congruence class is s-inde~sable. This fact, Tamura's theor~n, has been proved in [i], [2], [3] and [4]. In this note, semilattice congruences viewed frcm certain set-valued functions are handled and Tamura's theorem will be proved in a somewhat different way. Let C(S) be the set of semilattice congruences on S and F(S) the set of set-valued functions M on S each of which satisfies the following three conditions for any x, yES: (I) x E M(x) ~S. (2) y ~ M(x) implies M(y) _c M(x). (3) M(x) f% M(y) = M(xy). Then ( F(S), < ) is a partially ordered set where for