正则密码群并半群的两个等价刻画

Two Equivalent Descriptions of Regular Cryptogroups

借助同余和关系同态,证明了以下3条性质在完全正则半群S=(Y;Sα)上等价:(i)S是正则密码群并半群;(ii)■a∈S,等价关系ρa={(x,y)∈S×S:(axa)0=(aya)0}是S上的同余;(iii)■α,β∈Y,α≥β,存在关系同态Φα,β:Sα—→ρα,β 2Sβ,使得■a∈Sα,b∈Sβ,有ab=(aΦα,βb)b,且ba=b(aΦα,βb).

Abstract： By using congruences and relational homomorphisms, we prove that the following three state ments are equivalent on a completely regular semigroup S= （Y; Sa） ： （i） S is a regular cryptogronp; （ii） For any a∈ S, the equivalence ρa ={（x,y）∈S×S：（axa）^0=（aya）^0} is a congruence on S; （iii） For any α,β∈Y, with α≥β, there exists a relational homomorphism Φα,β：Sα→ρα,β2^Sβ such that for any a∈Sα,b∈Sβ, ab=（aΦα,β^b）b, and ba =b（aΦα,β^b）.