关于局部有限半群上的Brown定理的一个推广

An expansion of Brown's theorem on the locally finite semigroups

用不同的方法证明了T.C.Brown在[1]中证明的一个定理:设S和T是半群,:S→T是态射,如果T是局部有限的,且对每个幂等元e∈T,e-1是局部有限的,则S是局部有限的.并把它推广到强局部有限半群的情况,证明了如果T是强局部有限半群,有阶函数f,且对每个幂等元e∈T,e-1是强局部有限的,有同一个阶函数g,则S是强局部有限的,且有一个从f和g可算的阶函数.

In this paper we give new proofs of following theorem, due to T.C.Brown\: Let \%S\% and \%T\% be semigroups and let \%φ:S→T\% be a morphism. If \%T\% is locally finite and if for each idempotent e∈\%T\%,e\%φ\+\{-1\}\% is locally finite, then \%S\% is locally finite. Furthermore we expand it to the case for strongly locally finite semigroup, and prove the following theorem: if \%T\% is strongly locally finite with order function \%f\% and all e\%φ\+\{-1\}\%, where e∈\%T\% is idempotent, are strongly locally finite with a common order function \%g\%, then \%S\% is strongly locally finite and an order function for \%S\% can be effectively computed from \%f\% and \%g\%.