半群理论作业代做 MT5863代写 半群理论代写 数学作业代写
582MT5863 Semigroup theory: Problem sheet 3 半群理论作业代做 Binary relations and equivalences 3-1. Let X = {1, 2, 3, 4, 5, 6}, let ρ be the equivalence relation on X with equivalence classes {1,...
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半群理论代写 Let S be a semigroup, and let e, z, u ∈ S. Then: (i) e is a left identity if ex = x for all x ∈ S; (ii) e is a right identity if xe = x for all x ∈ S
Let S be a semigroup, and let e, z, u ∈ S. Then:
(i) e is a left identity if ex = x for all x ∈ S;
(ii) e is a right identity if xe = x for all x ∈ S;
(iii) z is a left zero if zx = z for all x ∈ S;
(iv) z is a right zero if xz = z for all x ∈ S;
(v) z is a zero if it is both a left zero and a right zero;
(vi) e is an idempotent if e2 = e.
(a) Suppose that S has a left identity e and a right identity f. Show that e = f and S has a 2-sided identity.
(b) Prove that if S has a zero element, then it is unique.
(c) Is it true that if a semigroup S has left zero and right zero, then they are equal and S has a zero element?
1-2. Prove that the size of the full transformation semigroup Tn is nn.
1-3. Prove that a mapping f ∈ Tn is a right zero if and only if it is a constant mapping. Does Tn have left zeros? Does it have a zero? Does Tn have an identity?
rank(fg) ≤ min(rank(f),rank(g)).
Find examples which show that both the equality and the strict inequality may occur.
1-7. Let G be a group and let a ∈ G. Then define
aG = { ag : g ∈ G } and Ga = { ga : g ∈ G }.
Prove that aG = Ga = G for all a ∈ G.
1-8.* Let S be a non-empty semigroup such that aS = Sa = S for all a ∈ S.
(a) If b ∈ S is arbitrary, then prove that there exists an element e ∈ S such that be = b.
(b) Prove that e is a right identity for S.
(c) In a similar way prove that S has a left identity too. Conclude that S is a monoid.
(d) Prove that S is a group.
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MT5863 Semigroup theory: Problem sheet 3 半群理论作业代做 Binary relations and equivalences 3-1. Let X = {1, 2, 3, 4, 5, 6}, let ρ be the equivalence relation on X with equivalence classes {1,...
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