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# 数学半群代写 半群理论课业代写 MT5863代写 半群理论代写

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## Green’s relations again, Simple semigroups

### Green’s relations again数学半群代写

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

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

7-4.

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

(b) Prove that there is an injective homomorphism from Pn to Tn+1.

(c) Prove that the following hold in Pn:

7-5. Let S be a semigroup and let e S be an idempotent. Suppose that Le, Re and De are the L -, R-, and Dclass of e, respectively. Prove that LeRe = De.

7-6. Let a, b be elements in a D-class D. Show that ab Ra Lb if and only if Rb La 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 Tn 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?

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