正则半群上的群同余关系

GROUP CONGRUENCE ON A REGULAR SEMIGROUP

通过引进半群的内酉子半群和正则半群的完全内酉子半群的概念，讨论了正则半群上的群同余与其完全内酉子半群之间的对应关系．

The concept of completely inner-unitary subsemigroup of a regular semigroup is introduced,and the relation between completely inner-unitary subsemigroup and group congruence on a regular semigroup is discussed. The following results are obtained:(1) Suppose that H is an inner-unitary subsemigroup of a regular semigroup S, then the principal congruence ρH= { (a,b) ∈SXS: H.. α=H.. b}= { (a,b) ∈S×S :a [-1]H =b[-1]H} = { (a,b) ∈ S ×S: Ha[-1]=Hb[-1] },and if a[-1] H ≠ Ф for any α∈S,then S/ρH is a group ;if α[-1]H= Ф for some α∈ S,then S/ρH is a 0-group.(2) Let S,H and ρH be as (1 ),then S/ρH is a group if and only if H is a completely inner-unitary subsemigroup of S (i. e. H is inner-unitary and E(H)=E(S)).(3) Suppose that ρ is a group congruence on a regular semigroup S,then Hρ= {a∈S : αρ be the identity of S/ρ}is a completely inner-unitary subsemigroup of S.(4) Suppose that S is a regular semigroup and C(S)= {ρS ×S : ρ be a group congruence on S }and U(S) = {H=S: H be a completely inner-unitary subsemigroup of S) },let Φ: C(S)→U(S), Hρ=Hρ for any ρ∈ C(S),Φ: U(S)→C(S),HΦ=ρH for any H ∈ U(S),then ρ(Φ ° Φ)=ρ for any ρ∈C(S),and H(Φ°Φ)=H for any H∈ U(S).