Some Properties On Strong Semilattice Of Inverse Semigroups

In this dissertation, we characterize the ralation of the representations on a strong semilattice of inverse semigroups and the representations on each inverse semigroup;besides,we give the sum of representation of inverse semigroups;finally, we discuss the ralation of the semidirect product on a strong semilattice of inverse semigroups and the semidirect product on each inverse semigroups. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries .In Chapter 2, we characterize the ralation of the faithful representations on a strong semilattice of inverse semigroups and the faithful representations on each inverse semigroups;besides, we give the sum of represention of inverse semigroups.Theorem 2.1.2 Let S = [Y;S_a, Φ_α,β] be a strong semilattice of inverse semigroups,Theni.e., if , we havewhere b = aΦ_α,βj ((?)β ≤ α).Theorem 2.1.5 Let S = [Y;S_a, Φ_α,β] be a strong semilattice of inverse semigroups , then S^W be a strong semilattice of S_α^W_α(α ∈ Y).Theorem 2.2.2 Let 5 = [Y;Sa, $a,3],5aa: [aa-l)[\Ea^[arla\(\Ea,5: S->${E),a>-> 6a,5a : [aa"1] —? [a"1^, e \-> a^xea.i.e., if Va 6 Sa(a G Y),e e [aa^1], we havej, x e Ep, (V/3 < a), where b = a$Q>^ (V/3 < a).Theorem 2.3.2 Let 5 = [Y;Sa;$Qi/J],0% : aSa^1 -> a^lSaa, x i-> a¹xa, 9: S->V{S),a^ 9a, 9a : aSa^x ->? a^lSa, x (->â–  a'^a. Then ^|Sq= Ja0(?(? U.)s,)-i.e., if Va G Sa(a eY),xe aSa'1, we have== x9a,x% x e aSa'1 f]S0, (V/3 < a),where b = a$a^ (V/3 < a).Theorem 2.4.7 Let 5 = [Y;5a,$Qi)3], the set P{S) of all subsets of S is a group under the operation of symmetric difference of subsets, i.e., Vj4, B G P(S)AoB = (A\JB)-(AnB),Letfa: Sa->J(P(Sa))-}s^ ff,ft : M?{s) -v M?(s);A >-+ As, f: S^J(P(S));s^ /? /,: Mi(s) ^ M2(s);A\-+As,where M?{s) = {A € P(Sa)\Ass^l =M2Q(s) - {A e P(5Q)|As-1s =Then /k=/?0(?(/k)s,). i.e., if Vs G SqjA € Mi(s), we havewhere t = s$a>J3 (V/3 < a).Theorem 2.4.10 Let 5 = [Y;Sa,$a,p] be a strong semilattice of inverse semigroups, then S* be a strong semilattice of S{°(oi e Y).In Chapter 3, we characterize the ralation of the multi-automorphism representations on a strong semilattice of inverse semigroups and the multi-automorphism representations on each inverse semigroup. The main results are given in follow .Theorem 3.1.5 Let S - [Y;Sa,$aj], the set P(S) of all subsets of S is a group under the operation of symmetric difference of subsets, i.e., if VA, B ï¿¡ P{S),A o B = {A U B) - {A D B). Let Q(P(S))is the set of all sunsets of multi-automorphism of P(S),Q(P(Sa))is the set of all sunsets of multi-automorphism of P{Sa),\/s € S, A,B e P{S),A = (jAj,B = {JB^Aj C SlrB% C Sa.,(i = 1,2,-?â–  ,n,j = 1,2,..- ,p,),g? € Q(P(SQ))(Va € Y,a{ € V, Then there is some set i7 which satisfy the following resultwhere g* e Q(P{Sai)), t3 = 5$Q,Qi, Va, e H.In chapter 4, we discuss the ralation of the semidirect product on a strong semilattice of inverse semigroups and the semidirect product on each inverse semigroups. The main results are given in follow.Theorem 4.7 Let S — (Y;Sa, $a,p) be the strong semilattice of inverse semigroups, suppose that Sa is isomorphic to the semidirect product of Ea by G, a € Y,where Ea is a semilattice,G is a group, E = \J Ea. If Va G Y, Ga is thesubgroup of group G,Va > j3 G Y, there is a homomorphism ï¿¡a^ : (7a -> Gd, it satisfies the following condition) (V C, e G 5Q),Then ï¿¡ xT G = {{e,g) G ï¿¡:o x Ga\E = [j Ea:Ga < G} is a semidirectproduct of E by G.Furtherly,^ = E xTG.