数学群代考 MT4003代写 Groups代写 数学考试代考
230MT4003 Groups 数学群代考 EXAM DURATION: 2 hours EXAM INSTRUCTIONS: Attempt ALL questions. The number in square brackets shows the maximum marks obtainable EXAM DURATION: 2 hours EX...
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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.
(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]
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]
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. [3]
(b) Show that every subsemigroup of In is a block group. [3]
(c) Prove that there exists a finite block group that cannot be embedded into any symmetric inverse monoid In, n ∈ N. [5]
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