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 [math]\displaystyle{ E }[/math] 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 := [math]\displaystyle{ \bigcup_{e,f\in E} }[/math] { 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 [math]\displaystyle{ E }[/math], the semilattice of idempotents of [math]\displaystyle{ T_E }[/math] is isomorphic to E.

Example

Let [math]\displaystyle{ E=\{0,1,2,...\} }[/math]. Then [math]\displaystyle{ E }[/math] is a semilattice under the usual ordering of the natural numbers ([math]\displaystyle{ 0 \lt 1 \lt 2 \lt ... }[/math]). The principal ideals of [math]\displaystyle{ E }[/math] are then [math]\displaystyle{ En=\{0,1,2,...,n\} }[/math] for all [math]\displaystyle{ n }[/math]. So, the principal ideals [math]\displaystyle{ Em }[/math] and [math]\displaystyle{ En }[/math] are isomorphic if and only if [math]\displaystyle{ m=n }[/math].

Thus [math]\displaystyle{ T_{n,n} }[/math] = {[math]\displaystyle{ 1_{En} }[/math]} where [math]\displaystyle{ 1_{En} }[/math] is the identity map from En to itself, and [math]\displaystyle{ T_{m,n}=\emptyset }[/math] if [math]\displaystyle{ m\not=n }[/math]. The semigroup product of [math]\displaystyle{ 1_{Em} }[/math] and [math]\displaystyle{ 1_{En} }[/math] is [math]\displaystyle{ 1_{E\operatorname{min} \{m, n\}} }[/math]. In this example, [math]\displaystyle{ T_E = \{1_{E0}, 1_{E1}, 1_{E2}, \ldots \} \cong E. }[/math]

References

• Howie, John M. (1995), Introduction to semigroup theory, Oxford: Oxford science publication .
• Mitchell, James D. (2011), Munn semigroups of semilattices of size at most 7, http://www-groups.mcs.st-andrews.ac.uk/~jamesm/munn/index.php .