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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.

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