数学半群代做 MT5863代写 半群理论作业代写 半群作业代写

MT5863 Semigroup theory: Problem sheet 10

Inverse semigroups again, Clifford semigroups

数学半群代做 Inverse semigroups 10-1. Let E be a partially ordered set, and let e, f ∈ E. We say that k ∈ E is a greatest lower bound for {e, f} if k ≤ e, k ≤ f

Inverse semigroups 数学半群代做

10-1.

Let E be a partially ordered set, and let e, f ∈ E. We say that k ∈ E is a greatest lower bound for {e, f} if k ≤ e, k ≤ f and, if g ≤ e and g ≤ f then g ≤ k. If a greatest lower bound exists then we write it e ∧ f.

(a) Suppose that E is a partially ordered set such that for any e, f ∈ E, the greatest lower bound e ∧ f exists. Show that defining a multiplication ef = e ∧ f makes E a semilattice (in the semigroup sense).

(b) Suppose now that E is a semilattice (in the semigroup sense) and say that e ≤ f if ef = e. Show that this is a partial order on E and that every pair of elements has a greatest lower bound with respect to this ordering. 数学半群代做

10-2. Let S be an inverse semigroup and let E be the semilattice of idempotents of S. Recall that the natural partial order on S is defined by a ≤ b if there is e ∈ E such that be = a.

(a) Show that ≤ is a partial order on S.

(b) Show that if a, b, c ∈ S with a ≤ b then ac ≤ bc and ca ≤ cb.

(c) Show that a ≤ b if and only if ba−1a = a.

Clifford semigroups 数学半群代做

10-3. Let S be an inverse semigroup and let E be the set of idempotents of S. Show that ae = ea for all a ∈ S and e ∈ E if and only if aa⁻¹ = a⁻¹a for all a ∈ S.

10-4. Let S be a semigroup. Show that S is completely regular if for every a ∈ S there is x ∈ S such that axa = a and ax = xa.

10-5. Let S be a completely regular semigroup and let θ : S → T be a surjective homomorphism. Show that T is a completely regular semigroup. Deduce that the homomorphic image of a Clifford semigroup is a Clifford semigroup.

10-6. Prove that a simple semigroup is completely simple if and only if it is completely regular.

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