数学半群代写 半群理论课业代写 MT5863代写 半群理论代写

MT5863 Semigroup theory: Problem sheet 7

Green’s relations again, Simple semigroups

数学半群代写 Green’s relations again 7-2. Let Dr be the D-class of the full transformation semigroup Tn (n ≥ r) consisting of all the mappings with rank r.

Green’s relations again 数学半群代写

7-2. Let D_r be the D-class of the full transformation semigroup T_n (n ≥ r) consisting of all the mappings with rank r. How many L -classes, R-classes and H -classes does D_r contain? What is the size of D_r?

7-3. Prove that the H -class H_f of a mapping f ∈ T_n is a group if and only if rank(f) = rank(f² ).

7-4.

(a) Prove that the monoid of all partial transformations P_n is regular.

(b) Prove that there is an injective homomorphism from P_n to T_n+1.

(c) Prove that the following hold in P_n:

7-5. Let S be a semigroup and let e ∈ S be an idempotent. Suppose that L_e, R_e and D_e are the L -, R-, and D–class of e, respectively. Prove that L_eR_e = De.

7-6. Let a, b be elements in a D-class D. Show that ab ∈ R_a ∩ L_b if and only if R_b ∩ L_a contains an idempotent.

7-7. Show that if S is a periodic semigroup then a D-class D consists of regular elements if and only if there are a, b ∈ D such that ab ∈ D.

Simple semigroups 数学半群代写

7-8. Prove that a subsemigroup S of T_n is simple if and only if rank(f) = rank(g) for all f, g ∈ S.

7-9. Let S be a monogenic semigroup (which we recall means that there is s ∈ S such that S = ⟨s⟩). Prove that S is simple if and only if S is a group. What about if S = ⟨a, b⟩, is it still true that S is simple if and only if S is a group?

7-10. Prove that in a finite simple semigroup S

ac = bc and ca = cb ⇒ a = b

for any a, b, c ∈ S.

Is it true that

ac = bc ⇒ a = b

for any a, b, c ∈ S?

更多代写：Cs澳洲网课托管  gre代考靠谱吗  英国fin金融学代考  Essay代写表语从句  Chapter Summary代写 半群理论课业代写

合作平台：essay代写 论文代写 写手招聘 英国留学生代写