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 :=  ⋃ e , f ∈ E  { T_e,f : (e, f) ∈ U }.

The semigroup's operation is composition of 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 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 E n = { 0 , 1 , 2 , . . . , n } for all n . So, the principal ideals E m and E n are isomorphic if and only if m = n .

Thus T n , n = { 1 E n } where 1 E n is the identity map from En to itself, and T m , n = ∅ if m ≠ n . In this example, T E = { 1 E 0 , 1 E 1 , 1 E 2 , … } ≅ E .