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 (1929âÂÂ2008).
Let be a semilattice.
1) For all e in E, we define Ee: = {i â E : i ⤠e} which is a principal ideal of E.
2) For all e, f in E, we define T<sub>e,f</sub> as the set of isomorphisms of Ee onto Ef.
3) The Munn semigroup of the semilattice E is defined as: T<sub>E</sub> := { T<sub>e,f</sub> : (e, f) â U }.
The semigroup's operation is composition of partial mappings. In fact, we can observe that T<sub>E</sub> â I<sub>E</sub> where I<sub>E</sub> 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 1<sub>Ee</sub>.
For every semilattice , the semilattice of idempotents of is isomorphic to E.
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,