令M为一有限幺半群,k是域。幺半群代数kM的理想I称为双理想,若I=k{ m−m′∈I|m,m′∈M }。kM的双理想集记为ℬℐ(kM),M的同余格记为C(M)。本文指出包含序下的偏序集ℬℐ(kM)在运算I∨J=I+J,I∧J=k{ m−m′∈I∩J|m,m′∈M }下构成一个格,并证明了ℬℐ(kM)与C(M)的格同构。进一步利用幺半群M上的同余和商空间kM/I的线性无关组给出双理性的刻画。Let M be a finite monoid and ka field. An ideal I of the monoid algebra kMis a bi-ideal if I=k{ m−m′∈I|m,m′∈M }. The set of bi-ideals of kMis denote by ℬℐ(kM)and the congruence lattice of M by C(M). In the paper we indicate that the partially ordered set ℬℐ(kM)ordered by inclusions is a lattice under the operations I∨J=I+J, I∧J=k{ m−m′∈I∩J|m,m′∈M }, and show that two lattices ℬℐ(kM)and C(M)are isomorphic. Furthermore, we characterize bi-ideals in terms of congruences on monoid M and linear independence lists in the quotient space kM/I.
