毕竟正则半群上的群同余

Group congruences on an eventually regular semigroup

设S是一个半群,a∈S.如果存在x∈S,使得x=xax,则称x为a的一个弱逆.用W(a)表示a的所有弱逆的集合.本文利用元素的弱逆给出了毕竟正则半群S的群同余的若干等价刻画及一个表示.通过S的w-自共轭的、闭的,全子半群H定义了S上的一个二元关系(a,b)∈ρH(?)((?)a’∈W(a),a'b∈H),证明了如果H是S的w-自共轭的、闭的全子半群,则ρH是S上的以H为核的群同余.反过来,如果ρ是S上的群同余,则kerρ是S的w-自共轭的,闭的全子半群,并且ρ=ρkerρ.

Let S be an eventually regular semigroup and α∈S. A weak inverse of S is an element x∈S such that x=xαx, denoted by W（α） the set of weak inverses of α. A representation and some characterizations of group congruences on the eventually regular semigroup are given by means of weak inverse. Define a relation on S： （α, b） ∈ ρH〈=〉 （ α′∈W（α）, α′b∈H）. It is shown that if H is a ω-self-conjugate, closed and full subsemigroup of S, then ρH is a group congruence on S and ker ρH=H. Conversely, if ρ is a group congruence on S, then ker ρ is a ω-self-conjugate, closed and full subsemigroup of S and ρ=ρkerρ.