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 $E$ 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'' := $\bigcup_$ .

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=\$. Then $E$ is a semilattice under the usual ordering of the natural numbers ($0 < 1 < 2 < ...$).
The principal ideals of $E$ are then $En=\$ for all $n$.
So, the principal ideals $Em$ and $En$ are isomorphic if and only if $m=n$.

Thus $T_$ = where $1_$ is the identity map from En to itself, and $T_=\emptyset$ if $m\not=n$. The semigroup product of $1_$ and $1_$ is $1_$.
In this example, $T_E = \ \cong E.$

References

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*{{citation, last=Mitchell, first=James D., title=Munn semigroups of semilattices of size at most 7, url=http://www-groups.mcs.st-andrews.ac.uk/~jamesm/munn/index.php, year=2011.

Semigroup theory