Munn semigroup explained

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: =  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 := 

c up _{e,f\in}  .

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

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} = 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.

References

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