有限幂零半群的构造

THE STRUCTURE OF FINITE NILPOTENT SEMIGROUPS

有零元的半群S称为幂零的,如果对于任意x∈S有正整数n使x^n=0,设S是有限幂零半群,则存在正整数门使得S^n={0}([1],P.81),适合此条件的最小正整数称为S的0-阶,记为0(S);S的非零元x的0-阶(记为o(x))是指适合下述条件的最小正整数n:S的任意n个包括x在內的元素的积为零;零元的o-阶规定为1. 以下假设S是n阶有限幂零半群.

In this paper, the author discusses the structure of finite nilpotent semigroups. The main result is:'A semigroup S is afinite nilpotent semigroup if and only if S A/L[τ_1, …,τ_n], in which A is a finite alphabet (the minimum generated set of S), L is a b-full finite subset of A (the structure set of S); τ_i is an equivalence of L_i (the structure component of S) satisfying the following conditions: (1)τ_i P_L (the syntactic congruence of L), (2)τ_1=L_1×L_1, (3)(α∈A, x∈ A^+) ατ_ix α=x; and L[τ_1,…,τ_n] is the congruence of A^+ generated by {τ_i}.'