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# 半群理论代考 MT5823代写 数学半群代考 数学考试代考

76

## MT5823 Semigroup theory

EXAM DURATION: 2 hours

EXAM INSTRUCTIONS: Attempt ALL questions.

The number in square brackets shows the maximum marks obtainable for that question or part-question.

PERMITTED MATERIALS: Non-programmable calculators

YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE INSTRUCTED TO DO SO.

A block group is a semigroup S such that for every s ∈ S there exists at most one t ∈ S where sts = s and tst = t.

### 1.半群理论代考

(a) State the definition of a regular element of a semigroup and of a regular semigroup. 

(b) Show that if s S is a regular element, then there exists t S such that sts = s and tst = t. 

(c) Prove that a block group is an inverse semigroup if and only if it is regular. 

(d) Suppose that S is a semigroup with commuting idempotents (i.e. if e, f S are idempotents, then ef = fe). Show that S is a block group. 

(e) Give an example of a semigroup that is not a block group. 

### 3.半群理论代考

Let T be the semigroup defined by the multiplication given by the following table:

You may use the fact that T is a semigroup without proof.

(a) Show that T = 〈x, y  . 

(b) Give the definition of a Rees congruence on a semigroup and of a Rees quotient of a semigroup. 

(c) Show that T is isomorphic to a Rees quotient of the subsemigroup S of T3 from Question 2.

[Hint: Show that there is a homomorphism from S to T using Question 2(e) and show that the kernel of this homomorphism is a Rees congruence.] 

### 4.

Let In denote the symmetric inverse monoid on the set {1, 2, . . . , n}.

(a) Show that if e, f In are idempotents, then ef = fe. 

(b) Show that every subsemigroup of In is a block group. 

(c) Prove that there exists a finite block group that cannot be embedded into any symmetric inverse monoid In, n ∈ N. 

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