Munn semigroup


In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice. Munn semigroups are named for the Scottish mathematician Walter Douglas Munn.

Construction's steps

Let be a semilattice.
1) For all e in E, we define Ee: = 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 := .
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, the semilattice of idempotents of is isomorphic to E.

Example

Let. Then is a semilattice under the usual ordering of the natural numbers.
The principal ideals of are then for all.
So, the principal ideals and are isomorphic if and only if .
Thus = where is the identity map from En to itself, and if. The semigroup product of and is.
In this example,