MT5863 Semigroups
半群代写 1. (a) Define a right zero element in a semigroup, and a right zero semigroup. [2](b) Denote by Tn the full transformation monoid on {1, . . . , n}.
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1. 半群代写
(a) Define a right zero element in a semigroup, and a right zero semigroup. [2]
(b) Denote by Tn the full transformation monoid on {1, . . . , n}. A mapping α ∈ Tn is said to be constant if there exists y ∈ {1, . . . , n} such that xα = y for all x ∈ {1, . . . , n}. Prove that a mapping α is a right zero of Tn if and only if it is a constant mapping. [2]
(c) Does Tn (n > 1) have left zeros? Justify your answer. [2]
(d) Prove that the set C ⊆ Tn of all constant mappings is an ideal, and that this ideal is minimal with respect to inclusion. [2]
(e) Prove that if D ⊆ Tn is a subsemigroup isomorphic to a semigroup of left zeros, then all the elements of D have the same image. [2]
(f) Exhibit an example of a subsemigroup E of T4 isomorphic to the left zero semigroup of order 4. Justify your answer. [2]
2.
Let S be the semigroup with 11 elements generated by two elements a, b whose right Cayley graph is shown in Figure 1.
(a) Compute the product a2b · ab2 in S. [1]
(b) List the R-classes of S. Justify your answer. [2]
(c) Is R a congruence relation on S? Justify your answer. [2]
(d) Compute and draw the left Cayley graph of S. Explain your method. [6]
(e) Which, if any, of the R-classes are ideals of S? Justify your answer. [2]
3. 半群代写
(a) Show that B(G, X) is a 0-simple semigroup. [2]
(b) Give the definition of an inverse semigroup and show that B(G, X) is an inverse semigroup. [2]
(c) Let H be a normal subgroup of G and let
ρH = {((x, g, y),(x, h, y)) | x, y ∈ X, gh−1 ∈ H},
which is an equivalence relation on B(G, X). Show that ρH is a congruence on B(G, X) and that
B(G, X)/ρH ≌ B(G/H, X). [4]
(d) Show that if σ is a congruence on B(G, X) and there are (x, g, y),(z, h, w) ∈ B(G, X) such that ((x, g, y),(z, h, w)) ∉σ then σ ⊆ ρG. [1]
(e) Show that if σ is a congruence on B(G, X) such that σ ⊆ ρG then there is a normal subgroup K ≤ G such that σ = ρK. [3]
(f) Show that if f : B(G, X) → T is a surjective homomorphism then T is either trivial or a Brandt semigroup. [2]
4. 半群代写
In this question let S be a regular semigroup.
(a) Show that each R-class R of S contains an idempotent e and that e is a left identity for R. [3]
(b) Let D be a D-class of S. For s ∈ D let Ls and Rs be the L– and R-classes of s, respectively. Prove that there is some d ∈ D such that LdRd = D. [3]
(c) Suppose S is such that aS = S for all a ∈ S. Show that for all x, s, t ∈ S, if xs = xt then s = t. [5]
