# 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 ${\displaystyle E}$ 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 ef in E, we define Te,f as the set of isomorphisms of Ee onto Ef.

3) The Munn semigroup of the semilattice E is defined as: TE := ${\displaystyle \bigcup _{e,f\in E}}$ { Te,f : (ef) ∈ U }.

The semigroup's operation is composition of mappings. In fact, we can observe that TE ⊆ IE where IE 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 1Ee.

## Theorem

For every semilattice ${\displaystyle E}$, the semilattice of idempotents of ${\displaystyle T_{E}}$ is isomorphic to E.

## Example

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

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, ${\displaystyle T_{E}=\{1_{E0},1_{E1},1_{E2},\ldots \}\cong E.}$