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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.