半群理论代写 MT5863代写 半群理论定义和基本属性代写

MT5863 Semigroup theory: Problem sheet 1

Definition and basic properties

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.

1-1. Let S be a semigroup. 半群理论代写

(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 T_n is nⁿ.

1-3. Prove that a mapping f ∈ T_n is a right zero if and only if it is a constant mapping. Does Tn have left zeros? Does it have a zero? Does T_n have an identity?

1-6. Prove that for any f, g ∈ T_n  半群理论代写

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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