MT5863 Semigroup theory: Problem sheet 4
Congruences and presentations
半群理论课业代写 4-1.Prove that every equivalence relation on a semigroup of left zeros is a congruence. Find an equivalence relation on a rectangular
Congruences 半群理论课业代写
4-1.
Prove that every equivalence relation on a semigroup of left zeros is a congruence. Find an equivalence relation on a rectangular band which is not a congruence.
4-2.
Let S be a semigroup, and let ρ be a congruence on S. Prove that if e ∈ S is an idempotent, then its equivalence class e/ρ is a subsemigroup of S, and e/ρ is an idempotent in the quotient S/ρ. Also, prove that if S is finite and x/ρ is an idempotent of S/ρ, then x/ρ contains an idempotent.
4-3.
Let S = I × Λ be a rectangular band where |I| = |Λ| = 2.
(a) Show that if ρ is a congruence on S and (1, 1)ρ(2, 2), then ρ = S × S;
(b) Describe the least congruence ρ on S such that (1, 1)ρ(1, 2).
(c) Prove that S has 4 distinct congruences, and describe the quotient of S by each of these congruences.
4-4. Let ρ and σ be congruences on a semigroup S such that ρ ⊆ σ. Prove that
σ/ρ = { (x/ρ, y/ρ) ∈ S/ρ × S/ρ : (x, y) ∈ σ }
is a congruence on S/ρ and that (S/ρ)/(σ/ρ) ≌ S/σ.
Presentations 半群理论课业代写
4-5. Let S be the semigroup defined by the presentation
〈a, b|a3 = a, b4 = b, a2b = ba 〉.
Prove that S has order at most 11. In fact, |S| = 11, and you can use this in the rest of the question without proof. Find the idempotents of S. Draw the right Cayley graph of S.
4-6. Let S be the semigroup defined by the presentation
〈a, b, 0|a2 = b2 = 0, aba = a, bab = b, 02 = 0, 0a = a0 = b0 = 0b = 0〉.
Prove that S has order 5. Write down the multiplication table for S. Draw the left and right Cayley graphs of S.
4-7. Let S be the semigroup defined by the presentation
〈a1, . . . , an | a1a2 = a1, a2a3 = a2, . . . , an−1an = an−1, ana1 = an〉.
Prove that aiaj = ai for any i and j, and so S is the semigroup of left zeros of order n.
4-8. Consider the monoid S defined by the presentation 〈x, y | xyx = 1〉. Prove that xy = yx holds in S. Prove that every element of S is equal to one of xi , yj , xyj (i ≥ 0, j ≥ 1). Find two integers which generate the additive (semi)group Z and satisfy 2p + q = 0. Prove that S ≌ Z.
