Congruences And Characters On π-Regular Semigroups

In this dissertation, we mainly describe some congruences and characters onGV-semigoups, in fact, we extend some results of completely regular semigroup toGV-semigroups.There are two chapters in this paper.In the first chapter, we investigate some characters on GV-semigroups.Firstly;wegive the equivalent condition ouÏ€-cryptogroup as follows:(1) S isÏ€-cryptic:(2) S is a band ofÏ€-groups:(3) S satisfies the identity r(ab)⁰ = r(r(a)⁰r(b)⁰)⁰.Secondly, we give some characters on GV-semigroups S=(?) S_a when theset of idempotents E(S) is subsemigroup. that are the characters of GV-orthodoxsemigroups:(1) for anyα∈Y, S_αis a nil-extension of rectangular group:(2) the set of idempotents E(S)is selfconjugate;(3) for any∈E(S), V(e)(?)E(S);(4) S satisfies the identity r(a)⁰r(b)⁰=r(r(a)⁰r(b)⁰)⁰;(5) for any a, b∈S, V(r(b))V(r(a)) (?)V(r(a)r(b)).Lastly, we discuss the orthodoxy and E-unity between S/ρand S whenρis anidempotent pure congruence. The last section discuss some characters on completelyarchimedean semigroups.In the second chapter, we deal with some congruences on GV-semigroups.In the first section we discuss the group congruence onÏ€-regular semigroups,it is theextension of kernel and trace on regular semigroups.First we give the definitionof congruence subsemigroup onÏ€-regular semigroup:if K is full,selfconjugate andunitary.Given such a congruence subsemigroup,we characterize group congruenceρ_kon S:(a, b)∈ρ_k(?)There exist x∈RegS with ax, bx∈K.The second section give the least group congruenceρon nil-extension of rect-angular groups:(a,b)∈ρ(?)There exist e∈E(S)with eae=ebe.Then we characterize the least Clifford-semigroup congruence on semilatticeof nil-extension of rectangular groups S by least group congruence on nil-extensionsof rectangular groups, that is,if S=(?) S_{α·ρα}is the least group congruence on S_α.then the relationρdefining as follows is the least Clifford-semigroup congruence onS:(a, b)∈ρ(?)There existα∈Y with a,b∈S_α, and (a,b)∈ρ_α.We also get the least Clifford-semigroup congruence on semilattices of nil-extensions of left groups and semilattices of nil-extensions of right groups.The last section characterize Clifford-semigroup congruence on GV-semigroupS=(?) S_αby giving the group congruence on S_α, The main result is:S=(?) S_αis a GV-semigroup,ρ_αis group congruence on S_αas the definitionof the first section,thenσ=<(?)ρ_α> is the Clifford-semigroup congruence.Ifρis Clifford-semigroup congruence on GV-semigroup S=(?) S_α, letρ_α=ρ|s_α, thenρ_αis group congruence on S_α,and <(?)ρ_α＞(?)ρ.Particularly,ρ=<(?)ρ_α>ifand only if aρb(?)aJ^*b.