The Bi-Ideals, Semidirect Products And Other Properties On Some Semigroups

In chapter 1, we mainly discuss bi-ideals.Maria.Maddalena Miccoli(Lecce)[19] discussed relations of bi-ideal sets on regular semigroups and orthogroups.GV-cryptic orthogroup is an important kind of semigroups.The relations between B(S), B(E),B(S/H^*),B(RegS) on Gv-cryptic orthogroup are unknown. In the chapter ,we give their relations and several properties of quasi B^*-pure semigroup. Yin Xunjuan depicted some∏- regular semigroups by ideals,two-sided principal ideals and proper ideals.We discussed the relations of bi-ideals and some∏-regular semigroups. Then the relations between∏- regular semigroups and ideals are better and richer.The main results are given as follows:Theorem1.2.3 Let S be a GV-cryptic orthogroup,then B(Reg(S))(?)B(E(S)).Theorem1.2.4 Let 5 be a GV-cryptic orthogroup,then B(S) is not isomorphic to B(S/H^*).But there is morphism from B(S) onto B(S/H^*).Theorem1.2.7 Let S be a GV-cryptic orthogroup.then E(?)S/H^*.Corollary 1.2.8 Let S be a GV-cryptic orthogroup.then B(E)(?)B(RegS)(?) B(S/H^*).Corollary1.2.9 Let S be a GV-cryptic orthogroup,then f : B(S)(?)B(S/H^*) (?)B(RegS)(?)B(E)(?)E,where f is a morphism .Corollary 1.2.11 Let S be a∏-regular semigroup,if every bi-ideal is∏-regular then 5 is a completely∏-regular semigroup.Theorem1.2.13 Let S be a quasiB^*-pure semigroup,then the following conditions are true:(1) For every a∈S, there is n∈N, then aⁿS = aⁿ⁺¹S. where i∈N.(2) S is a completely∏-regular semigroup.Theorem1.3.4 Let S be a semigroup,S is a∏-regular semigroup (?) every (proper) bi-ideal of S is a∏-regular semigroup.Theorem1.3.5 Let S be a semigroup.S is a completely∏-regular semigroup (?) every (proper) bi-ideal of 5 is a completely∏-regular semigroup.Theorem1.3.7 Let S be a semigroup,then:(1) S is a left∏- inverse semigroup (?) every (proper) bi-ideal of 5 is a left∏- inverse semigroup;(2) S is a right∏- inverse semigroup (?) every (proper) bi-ideal of 5 is a right∏- inverse semigroup;(3) 5 is a∏-inverse semigroup (?) every (proper) bi-ideal of S is a∏-inverse semigroup:Theorem1.3.11 Let 5 be a semigroup,then:(1) S is a∏- orthodox semigroup (?) every bi-ideal of 5 is an orthodox∏-inverse semigroup;(2) S is a strongly∏-inverse semigroup (?) every bi-ideal of 5 is a strongly∏-inverse semigroup;Corollary1.3.12 S is a GV-cryptic orthogroup (?) every bi-ideal of S is a GV -cryptic orthogroup.Corollary1.3.14 S is a∏- orthogroup with bands of idempotents of type B ,where B is any of the types of band classified by M.Petrich in [23] if and only if every bi-ideal of S is a∏-orthogroup with bands of idempotents of type B .Corollary1.3.15 S is a GV-orthogroup with bands of idempotents of type B ,where B is any of the types of band classified by M.Petrich in [23] if and only if every bi-ideal of S is a GV-orthogroup with bands of idempotents of type B .In chapter 2, we mainly discuss semidirect products.Many authors in [5]-[18],[36]-[40]have discussed many kinds of semidirect products on regular semigroups (or monoids)and∏- regular semigroups. Saito [12]disscussed semidirect product of inverse monoids.Zhang Ronghua [17]depicted semidirect product of inverse semigroups by means of S and T^e.We improve the obove results . We get the result that the semidirect product of inverse semigroups has the same properties with that of inverse monoids by that every element has only one inverse element .The main results are given as follows:Theorem 2.2.1 S.T are semigroups.The semidirect product S×_αT is an inverse semigroup,then the following conditions are true:(1) For any e∈E(S),t∈T,if t^et = t .then t^e = t.(2) S is an inverse semigroup ,T is a regular semigroup,and T^e is an inverse subsemigroup of T for any e∈E(S) .(3) (s. t)∈E(S×_αT) (?) s∈E(S), t∈E(T).(4) For any e∈E(S).u∈E(T),then u^e=u.(5) T is an inverse semigroup.(6) For any e∈E(S),t∈T. then t^e= t.Theorem 2.2.2 S.T are semigroups.The semidirect product S×_αT is an inverse semigroup if and only if the following conditions are true:(1) S.T are inverse semigroups .(2) For any e∈E(S),t∈T, then t^e = t.Corollary 2.2.6 5, Tare semigroups.Then the wreath product SW_XT is an inverse semigroup if and only if the following conditions are true:(1) S. T are inverse semigroups .(2) For any e∈E(S), f∈T^X,then f^e = f. Theorem 2.3.2 S,T are semigroups. S×_αT is aσ- inverse semigroup if and only if the following conditions are true:(1) S.T areσ-inverse semigroups .(2) For any e∈E(S),t∈T, then t^e = t.(3) For any s∈S,t∈T,there are u∈E(T), m∈N,then (tu)^s(m) = (u^st)(s(m)).Theorem 2.3.3 S, Tare semigroups.Then the wreath product SW_XT is aσ-inverse semigroup if and only if the following conditions are true:(1) S, T_X areσ-inverse semigroups .(2) S, T^X are regular semigroups,and for any e∈E(S), f∈E(T^X). we have f^e = f.(3) For any s∈S,f∈T,there are m∈N,(e, f₁)∈E(SW_XT).then(se)^m =(es)^m, (f^ef₁)^s(m)f^ef₁=(f₁^sf)^es(m)f₁^sf.Especially,if s∈E(S),then (ff₁)^m= (f₁f)^m.Lemma 2.3.4 T is a semigroup,then T is aσ-inverse semigroup (?) T^X is aσ-inverse semigroup (?) (T^X)^e is aσ-inverse semigroup.Theorem 2.3.5 5. T are semigroups.Then the wreath product SW_XT is aσ-inverse semigroup if and only if the following conditions are true:(1) S,T areσ- inverse semigroups.(2) S, T are regular semigroups , for any e∈E(S), f∈E(T^X). then f^e= f.(3) For any s∈S,f∈T,there are m∈N,(e, f₁) G E(SW_XT) and we have(se)^m = (es)^m, (ff₁)^s(m)ff₁= (f₁^sf)^s(m)f₁^sf.