幺半群的强半格上的Rees矩阵半群的平移壳

The Translational Hull of the Rees Matrix Semigroup over the Strong Semilattice of Monoids

设含幺元的半群A是幺半群A_e的半格,其中A的幺元为1_A,A_e的幺元为e,所有幺元e的集合为E（A）,则对于幺半群A上的Rees矩阵半群S和幺半群A_e上的Rees矩阵半群S_e,以下五个条件是等价的：（1）任意的e∈E（A）,a∈A,有ae=ea;（2）A是幺半群A_e的强半格;（3）S是S_e的强半格;（4）A的平移壳和A_e的平移壳的强半格同构;（5）S的平移壳和S_e的平移壳的强半格同构.

Let A be a monoid and A be the semilattice of monoids {Ae} with tile identity of A is 1A and the identity of Ae is e. Let E（A） be the set of all the identities e of Ae. Let S be the Rees semigroup over A and Se be the Rees semigroup over Ae. Then the the following five conditions are equivalent：（1）For all e C E（A）, a E A,ae = ea;（2）A is the strong semilattice of the monoids {Ae};（3）S is the strong semilattice of {Se}; （4） The translational hull of A and the strong semilattice of the translational hulls of {Ae} are isomorphic; （5）The translational hull of S and the strong semilattice of the translational hulls of {Se} are isomorphic.