Munn semigroup

In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Douglas Munn (1929–2008).^[1]

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

Thus ${\displaystyle T_{n,n}}$ = {${\displaystyle 1_{En}}$} where ${\displaystyle 1_{En}}$ is the identity map from En to itself, and ${\displaystyle T_{m,n}=\emptyset }$ if ${\displaystyle m\not =n}$. In this example, ${\displaystyle T_{E}=\{1_{E0},1_{E1},1_{E2},\ldots \}\cong E.}$

References

1. O'Connor, John J.; Robertson, Edmund F., "Walter Douglas Munn", MacTutor History of Mathematics archive, University of St Andrews.

• 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.