Two Equivalent Descriptions of Regular Cryptogroups
Wang Zheng-pa · Journal of Southwest University · 2013
By using congruences and relational homomorphisms,we prove that the following three statements are equivalent on a completely regular semigroup S =(Y; Sα) :(i)S is a regular cryptogroup; (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,β 2Sβ such that for any a∈Sα,b∈Sβ,ab =(aΦα,βb) b,and ba = b(aΦα,βb).