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数学半群代做 MT5863代写 半群理论代写

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MT5863 Semigroup theory: Problem sheet 8

 (0-)Simple semigroups, Rees (0-)matrix semigroups

数学半群代做 Simple semigroups 8-1. Let S be a finite 0-simple semigroup. Prove thatabc = 0 ⇒ ab = 0 or bc = 0 for any a, b, c ∈ S. Is it true that ab = 0

Simple semigroups   数学半群代做

8-1. Let S be a finite 0-simple semigroup. Prove that

abc = 0 ab = 0 or bc = 0

for any a, b, c S.

Is it true that ab = 0 implies that a = 0 or b = 0?

8-2. A sequence of (two sided) ideals S1 S2 ⊃ · · · ⊃ Sm is called a principal series for a semigroup S if S1 = S, Sm is the minimum ideal of S and there is no ideal I such that Si I Si+1.

Show that the factors S1/S2, S2/S3, . . . Sm1/Sm are isomorphic to the principal factors of the semigroup which correspond to J -classes which are not the minimal ideal of the semigroup. Deduce that any two principal series have isomorphic factors.

 

Rees matrix semigroups    数学半群代做

 

 

8-7. Consider the monoid S defined by the presentation

a, b | a2 = 1, b3 = b, bab = b.

Using the standard methods one may show that |S| = 10 and

S = {1, a, b, b2 , ba, b2a, ab, ab2 , aba, ab2a}.

Find the L – classes and R– classes of S. Prove that M = S\{1, a} is the minimal two-sided ideal of S. Find a Rees matrix semigroup isomorphic to M.

8-8. Let S be a Rees matrix semigroup M[T; I,Λ; P] over a semigroup T. Prove that an element (i, t, λ) S is regular if and only if there are j I and µ Λ such that the set pλjT pµi contains an inverse of t.

8-9. Prove that a Rees matrix semigroup S = M[T; I,Λ; P] over a semigroup T is simple if and only if T is simple.

数学半群代做
数学半群代做

 

 

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