半群TE(X)中的一个同余

设ＦＸ表示集合上的全变换半群，Ｃｏｎ（Ｓ）表示半群Ｓ上的同余格，对Ｘ上任一非平凡等价关系Ｅ，令ＴＥ（Ｘ）＝（ｆ∈ＦＸ：Ａ↓（ａ，ｂ）∈Ｅ，（ｆ（ａ），ｆ（ｂ））∈Ｅ），据「４」，ＴＥ（Ｘ）构成一个α半群，且Ｃｏｎ（ＴＥ（Ｘ））可以 三个互不相交的完全子格，其中的一个为「Ｃ（Ｅ）〈Ｃａ（Ｅ）」，本文 ＴＥ（Ｘ）的同余τ，并证明了当Ｅ为单等价关系时，τ是「Ｃ（Ｅ），Ｃａ（Ｅ）」中的唯一原子。

A congruence on TE (X)

(fx) denotes the full transfomation semigroup on a set X and Con(S) denotes the complete lattice of all congruences on a senigroup S. For a nontrivial equivalence E on X, let TE(X) = {f∈(fx): (a, b) ∈E, (f( a),f( b ) ∈ E}. As we have seen in 4], TE (X) forms an α-semigroup and Con( TE (X) ) can be decomposed into three disjoint complete sublattices, one of which is C(E), Cα (E) ]. In this paper we find out a congruence τ on TE (X) which is contained in C ( E ), Cα ( E ) ] and we show that whenever E is a simple equivalence relation, τ is a unique atom of the complete sublattice C(E), Cα (E) ] of Con( TE (X) ).