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# 数学半群代做 MT5863代写 半群理论作业代写 半群作业代写

164

## Inverse semigroups again, Clifford semigroups

### 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 aa1 = a1a 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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