半群理论代考 MT5823代写 数学半群代考 数学考试代考

MT5823 Semigroup theory

半群理论代考 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. (a) State the definition of

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.

Your answers should contain the full working required to justify your solutions.

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. [1]

(b) Show that if s ∈ S is a regular element, then there exists t ∈ S such that sts = s and tst = t. [3]

(c) Prove that a block group is an inverse semigroup if and only if it is regular. [3]

(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. [6]

(e) Give an example of a semigroup that is not a block group. [2]

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 〉 . [1]

(b) Give the definition of a Rees congruence on a semigroup and of a Rees quotient of a semigroup. [1]

(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.] [5]

4.

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

(a) Show that if e, f ∈ I_n are idempotents, then ef = fe. [3]

(b) Show that every subsemigroup of I_n is a block group. [3]

(c) Prove that there exists a finite block group that cannot be embedded into any symmetric inverse monoid I_n, n ∈ N. [5]

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