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).[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 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 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 ${\displaystyle E}$, 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 ${\displaystyle m\not =n}$. The semigroup product of ${\displaystyle 1_{Em}}$ and ${\displaystyle 1_{En}}$ is ${\displaystyle 1_{E\operatorname {min} \{m,n\}}}$. 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.