Munn semigroup

Construction's steps

Let E be a semilattice.

1. For all e in E, we define Ee: = {i ∈ E : i ≤ e} which is a principal ideal of E.

2. For all e, f in E, we define T**e,f as the set of isomorphisms of Ee onto Ef.

3. The Munn semigroup of the semilattice E is defined as: T**E := \bigcup_{e,f\in E} { T**e,f : (e, f) ∈ U }.

The semigroup's operation is composition of partial mappings. In fact, we can observe that T**E ⊆ I**E where I**E is the symmetric inverse semigroup because all isomorphisms are partial one-one maps from subsets of E onto subsets of E.

The idempotents of the Munn semigroup are the identity maps 1Ee.

Theorem

For every semilattice E, the semilattice of idempotents of T_E is isomorphic to E.

Example

Let E={0,1,2,...}. Then E is a semilattice under the usual ordering of the natural numbers (0 ). The principal ideals of E are then En={0,1,2,...,n} for all n. So, the principal ideals Em and En are isomorphic if and only if m=n.

Thus T_{n,n} = {1_{En}} where 1_{En} is the identity map from En to itself, and T_{m,n}=\emptyset if m\not=n. The semigroup product of 1_{Em} and 1_{En} is 1_{E\operatorname{min} {m, n}}. In this example, T_E = {1_{E0}, 1_{E1}, 1_{E2}, \ldots } \cong E.

References

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References

1. "Walter Douglas Munn".