有限幺半群代数的双理想刻画

Characterizations of Bi-Ideals of Finite Monoid Algebras

摘要: 令M为一有限幺半群，$\mathbb{k}$ 是域。幺半群代数$\mathbb{k}M$ 的理想I称为双理想，若$I=\mathbb{k}\left\{m-m^{\prime }\in I|m,m^{\prime }\in M\right\}$ 。$\mathbb{k}M$ 的双理想集记为$ℬℐ\left(\mathbb{k}M\right)$ ，M的同余格记为$\mathcal{C}\left(M\right)$ 。本文指出包含序下的偏序集$ℬℐ\left(\mathbb{k}M\right)$ 在运算$I\vee J=I+J$ ，$I\wedge J=\mathbb{k}\left\{m-m^{\prime }\in I\cap J|m,m^{\prime }\in M\right\}$ 下构成一个格,并证明了$ℬℐ\left(\mathbb{k}M\right)$ 与$\mathcal{C}\left(M\right)$ 的格同构。进一步利用幺半群M上的同余和商空间$\mathbb{k}M/I$ 的线性无关组给出双理性的刻画。

Abstract: Let M be a finite monoid and $\mathbb{k}$ a field. An ideal I of the monoid algebra $\mathbb{k}M$ is a bi-ideal if $I=\mathbb{k}\left\{m-m^{\prime }\in I|m,m^{\prime }\in M\right\}$ . The set of bi-ideals of $\mathbb{k}M$ is denote by $ℬℐ\left(\mathbb{k}M\right)$ and the congruence lattice of M by $\mathcal{C}\left(M\right)$ . In the paper we indicate that the partially ordered set $ℬℐ\left(\mathbb{k}M\right)$ ordered by inclusions is a lattice under the operations $I\vee J=I+J$ , $I\wedge J=\mathbb{k}\left\{m-m^{\prime }\in I\cap J|m,m^{\prime }\in M\right\}$ , and show that two lattices $ℬℐ\left(\mathbb{k}M\right)$ and $\mathcal{C}\left(M\right)$ are isomorphic. Furthermore, we characterize bi-ideals in terms of congruences on monoid M and linear independence lists in the quotient space $\mathbb{k}M/I$ .