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 formula_1 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" := formula_2 { "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 1"Ee".
Theorem.
For every semilattice formula_1, the semilattice of idempotents of formula_4 is isomorphic to E.
Example.
Let formula_5. Then formula_1 is a semilattice under the usual ordering of the natural numbers (formula_7).
The principal ideals of formula_1 are then formula_9 for all formula_10. 
So, the principal ideals formula_11 and formula_12 are isomorphic if and only if formula_13.
Thus formula_14 = {formula_15} where formula_15 is the identity map from En to itself, and formula_17 if formula_18. The semigroup product of formula_19 and formula_15 is formula_21.
In this example, formula_22