### - Art Gallery -

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 E {\displaystyle E} 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 Te,f as the set of isomorphisms of Ee onto Ef.

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

The semigroup's operation is composition of partial 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 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 {\displaystyle m=n} 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

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.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index