Munn semigroup

In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Walter Douglas Munn (1929–2008).

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 Te,f as the set of isomorphisms of Ee onto Ef.

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

The semigroup's operation is composition of partial mappings. In fact, we can observe that TE ⊆ IE where IE 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 ($$0 < 1 < 2 < ...$$). 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, $$T_E = \{1_{E0}, 1_{E1}, 1_{E2}, \ldots \} \cong E.$$