Congruences On Some ∏-Regular Semigroups

In this dissertation, we mainly describe congruences on some Ï€-regular semigroups which all of regular elements can't form subsemigroups. The main ideal is to extend the concepts of kernel and trace. We give the concept of congruence pair, adding up to some conditions . Finally, we find out a bijection between congruence pairs and congruences ; In addition, we give and proof the least Clifford semigroup congruence , Clifford semigroup congruence and quasi-C-semigroup congruence on some 7r-regular semigroups. There are three chapters.In the first chapter, we deal with the r-semiprime Clifford semigroup congruence on the semilattice of nil-extensions of rectangular groups. Firstly, we give the definition of r-semiprime Clifford semigroup congruence pair (ζ, K) on the semilattice of nil-extensions of rectangular groups S, composed of a normal subsemigroup K of S and a semilattice congruence ζ on < E(S) >. K and ζ satisfy the following conditions ∨ a,b ∈ S, x ∈ K, e∈ E(S),(A) ea ∈ K, (r(o)⁰,e)∈ζ=>a∈K.(B) axb ∈ K, (r(ab)⁰,r(x)⁰) ∈ζ=ab∈ K.(C) r(a)r(b)^-1 ∈ K ==> V c∈ S, r(ca)r(cb)^-1∈ K, r(ac)r(bc)^-1∈ K.Given such a pair (ζ, K), we define a relation ρ(ζyK) on S by(a, b) ∈ ρ(ζ,K) = r(a)^-1r(b) ∈ K and (r(a)⁰,r(b)⁰)∈ ζ.Moreover, we show that (htrρ, kerρ) is r-semiprime Clifford semigroup congruence pair on the semilattice of nil-extensions of rectangular groups S and that ρ = ρ(htrρ,kerρ) for all r-semiprime Clifford semigroup congruence p on it. So we have this...