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: = {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 := $\bigcup _{e,f\in E}$ { 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 $E$, the semilattice of idempotents of $T_E$ is isomorphic to E.

Example

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

Thus $T_{n,n}$ = {$1_{En}$} where $1_{En}$ is the identity map from En to itself, and $T_{m,n}=\mathrm{\varnothing }$ if $m=n$. The semigroup product of $1_{Em}$ and $1_{En}$ is $1_{E\min \{m,n\}}$. In this example, $T_E=\{1_{E0},1_{E1},1_{E2},\dots \}\cong E.$

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