MT5863 Semigroup theory: Problem sheet 2
半群理论代做 Rectangular bands Let I and Λ be two sets. Define multiplication on the set S = I × Λ = { (i, λ) : i ∈ I, λ ∈ Λ } by (i, λ)(j, µ) = (i, µ).
Rectangular bands, cancellative semigroups, subsemigroups, monogenic semigroups, and idempotents
Rectangular bands
Let I and Λ be two sets. Define multiplication on the set S = I × Λ = { (i, λ) : i ∈ I, λ ∈ Λ } by (i, λ)(j, µ) = (i, µ). Such a semigroup is called a rectangular band.
2-1. Prove that a rectangular band S is a semigroup in which every element is idempotent. Also prove that xyz = xz for any x, y, z ∈ S.
2-2. Prove that a rectangular band has a left zero if and only if |Λ| = 1, in which case every element of S is a left zero.
Cancellative semigroups 半群理论代做
A semigroup is called cancellative if ax = ay ⇒ x = y and xa = ya ⇒ x = y for all a, x, y ∈ S.
2-3. Let e, a be elements of a cancellative semigroup S such that ea = a. Prove that e is an idempotent. Prove that e is the identity of S.
2-4. Does there exist a cancellative semigroup without an identity element?
2-5. Prove that every finite non-empty cancellative semigroup is a monoid. Can you find an example of an infinite cancellative semigroup without identity that is not free?
Monogenic semigroups and idempotents 半群理论代做
A semigroup is monogenic if it is generated by a single element.
2-9. If S is a finite monogenic semigroup, then prove that there exist m, r > 0 such that am+r = am, and that there is an idempotent power of a.
2-10. Prove that every finite semigroup contains an idempotent.
2-11. Does there exist a finite semigroup with exactly one idempotent? Does there exist an infinite semigroup without idempotents?
Further problems 半群理论代做
2-12.* Let N×N denote the set { (x, y) : x, y ∈ N }. Prove that N×N under the operation (x, y) + (z, t) = (x+z, y +t) forms a semigroup. Is N × N finitely generated?
2-13.* Prove that the rectangular band I×Λ can be generated using max(|I|, |Λ|) elements. Is there a smaller generating set?
2-14.* Let S be an infinite semigroup with the property that every countable subset is contained in a monogenic subsemigroup. Prove that S is isomorphic to the natural numbers under addition.
