Semidirect Products And Congruences Of Some Semigroups

In this dissertation, we mainly discuss semidirect products of GV-semigvoups,GV-inverse semigroups, a semilattices of nil-extensions of left(resp.right) groups,a semi-lattices of nil-extensions of rectangular groups.It is under the condition that semigroups have no identity elements that we get these results.We describe the closeness of semidirect products of these semigroups,Unlike the discussions of monoids,we describe the properties of semidirect products in virtue of S and T^e,the subsemigroup of T,not in virtue of S and T.The main results are given in follow:In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we describe the necessary and sufficient conditions for the semidirect products of S and T to be GV-semigroups and GV-inverse semigroups.The main results are given in follow.Theorem 2.1.2 Let S,T be two semigroups,α : S→ End(T),s → α(s) be a given homomorphism,then the semidirect products S ×_αT is a GV-semigroups if and only if:(i) for every e ∈ E(S), S and T^e are GV- semigroups,where T^e = {t^e|t ∈ T};(ii) for every s ∈ S, t ∈ T,there exists m ∈ Z⁺,such that s^m ∈ Reg(S),and t^[s(m)] ∈ (t^[s(m)])^s1smTt^[s(m)],where s₁ ∈ V(s^m);iii) for every s G Reg{S),t G T,if t G iSiaTSl8i,where si G V(s),then 3^ G T.such that t = {tst)s^ltfltst.Theorem 2.2.2 Let 5, T be two semigroups,a;: 5 —> End(T),s h-> a(s) be a given homomorphism,then the semidirect products 5 xa T is a GV-inverse semigroups if and only if:i) for every e G E(S),S and Te are GV-inverse semigroups,where Te — {te\t G T};ii) for every e G ï¿¡"(5), t G T,if tet = t.then te = t;iii) for every s € S.t ï¿¡ T,there exists m G Z+,such that sTO G /?e5r(5),and ^(m)i € (t[*(m)])*i*raTt[s(m)l,where sl G V'(sTO);iv) for every s G Reg(S),t G T,if < G islST5lSi,where si G F(s),then 3tx G T,such that t = (t't)8"1^"1^;v) for every e, f G E(S),u,v G T,if ueit = u^v^v — v,then 3n G ^+,such that(w/w)[(e/)(n)] = (ueu)[(/e)(n)]? End(T),s ?-> a(s) be a given homomorphism,then the semidirect products SxaT is a semilattices of nil-extensions of left groups if and only if:1) for every e G E(S),S and Te are semilattices of nil-extensions of left groups ,where Te = {te\t G T};2) for every e G E(S),t G T,if ft = t,then te = t\3) for every s € S,t G T,there exists m G Z+,such that sm G i?e^(S'),andt[s{m)\ € (ti?(?)i)*-^mT^s(m^;where sx G V(sm);4) for every s € Reg(S),t € T,if * G ï¿¡SlsTSl%then (M)5 = tt^where sx GV{s)AleT]5) for every e, f G E(S),u,v G T,if ueu = u,v^v — i',then 3n G Z+.such that(u/eveu)[(e/e)(n)] = (u/v)[(e/)(n)]_Theorem 3.2.2 Let 5, T be two semigroups,a : S —>â–  End(T),s i-> a(s) be a given homomorphism,then the semidirect products SxaT is a semilattices of nil-extensions of right groups if and only if:1) for every e G E(S), S and Te are semilattices of nil-extensions of right groups ,where Te = {ie|t G T};2) for every s G S, t G T,3m G Z+,such that sm G Reg(S), R t[s(m)1 G3) for every s G Reg{S),te T,iit G tsisrsist,where si G V'(s),then 3^ G T,such that t = (t'ty'tf'tH;4) for every e, / G E(S),u,v G T,if uen = n, u^u = v,then 3n G Z+,such that (u/c?eu)Ke/c)(n)] = (weu)K/6XnM.In chapter 4,we describe the necessary and sufficient conditions for the semidirectproducts of 5 and T to be a semilattices of nil-extensions of rectangular groups .The main results are given in follow.Theorem 4.1.2 Let S,T be two semigroups,a : 5 —>â–  End(T),s h-> a(s) be a given homomorphism,then the semidirect products S xQ T is a semilattices of nil-extensions of rectangular groups if and only if:1) for every e G E(S), S and Te are semilattices of nil-extentions of rectangular groups,where Te = {te\t G T};2) for every s G S,t G T,3m € Z+,such that sm G Reg(S), and tl*(mH G^[s(m)])SlS-T^(m)])Where Si e V(sm);3) for every s G Reg(S),t G T.if i G ï¿¡slSTSl%where Si G V(s),then 3?i G T,such that t = (ttt)*^lt81¹?t;4) for every e, f E E(S),u,v G T,if ueu — u,vfv = v,then 3n G Z+,such thatThese results make the area of research about semidirect products not to be confied to monoids .Thus the application of semidirect products as an instrument of studing semigroups is extend.