关于变换半群的子群结构

本文主要讨论变换半群的子群的性质和结构,得到的主要结果是:定理1 设B是A的一个非空子集,H是M(B)的一个子群,则有M(A)的子群G使得G_B=H且G与G_B同构。定理2 (1)设G是M(A)的一个子群,e是G的单位元,则G是M(A)的一个极大子群当且仅当G_Ae=∑_(Ae)。(2)M(A)的任何两个不同的极大子群之交是空集。

ON THE STRUCTURE OF SUBGROUPS OF TRANSFORMATION SEMIGROUPS

Let A be a nonempty set and M(A) be the set of all transformations on A.For u, v∈M(A), we define the compsition as usual: a~(uv)= (a~(uv)). for all a∈A,then M(A) is a semigroup with this compsition.Let B be a subset of A and u∈M(A), u_B denotes the restriction of u on B. Let G M(A), G_B={u_B|u∈G}. ∑_B denotes the set of all invertible tran- sformations on B.The main results of this paper is Theorem 1 Let B be a subset of A and H be a subgroup of M(B), then there is a subgrop G of M(A) such that G_B=H, G≌G_B. Teorem 2(1) Let G be a subgroup of M (A) with unit e, then G is a maximal subgroup of M (A) if and only if G_(Ae)= ∑_(Ae), (2) Let G, K be maximal subgroups of M (A), if G≠K, then G∩K=φ.