MT5823 Semigroups
数学半群代考 1. (a) Show that S has 6 elements. [4] (b) Draw the left and the right Cayley graphs of S. [4] (c) Find the R-, L -, and D-classes of S.
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.
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1. 数学半群代考
(a) Show that S has 6 elements. [4]
(b) Draw the left and the right Cayley graphs of S. [4]
(c) Find the R-, L -, and D-classes of S. Draw the partial order of the D-classes of S. [4]
(d) Show that S has 3 two-sided ideals and describe the Rees quotient of S by each of these ideals. [5]
(e) State the definitions of:
(i) a regular semigroup; [1]
(ii) the inverse of an element in a semigroup; [1]
(iii) an inverse semigroup. [1]
(f) Is S a regular semigroup? Is S an inverse semigroup? [3]
(g) Prove that an element of a semigroup is regular if and only if it has an inverse. [2]
2. 数学半群代考
A semigroup is called a rectangular group if it is isomorphic to a direct product of a group and a rectangular band. In other words, the semigroup S is a rectangular group if S ≌ G × R = {(g, r) : g ∈ G, r ∈ R} where G is a group and R is a rectangular band, with multiplication defined by
(g, r)(h, s) = (gh, rs).
A semigroup is orthodox if its idempotents form a subsemigroup.
Let G denote a finite group and R denote a finite rectangular band.
(a) State the definition of a rectangular band. [1]
(b) Let e denote the identity of G. Prove that {e} × R is the set of idempotents in G × R. Deduce that G × R is an orthodox semigroup. [3]
(c) State the definition of a simple semigroup. [1]
(d) Show that the rectangular group G × R is a simple semigroup. [4]
(e) State the Rees Theorem without giving a proof. [2]
(g) Show that for every i ∈ I and λ ∈ Λ there exist qi , rλ ∈ G such that pλ,i = rλqi . [5]
(h) Let I × Λ be a rectangular band, and denote the direct product of G and the rectangular band I × Λ by G × (I × Λ). Prove that ∅ : S → G × (I × Λ) defined by
(i, g, λ)∅ = (qigrλ,(i, λ))
is an isomorphism. Deduce that S is a rectangular group. [4]
(i) Show that a finite semigroup is simple and orthodox if and only if it is a rectangular group.
