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Munn semigroup

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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

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Let 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 ef 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 :=  { Te,f : (ef) ∈ 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

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For every semilattice , the semilattice of idempotents of is isomorphic to E.

Example

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Let . Then is a semilattice under the usual ordering of the natural numbers (). The principal ideals of are then for all . So, the principal ideals and are isomorphic if and only if .

Thus = {} where is the identity map from En to itself, and if . The semigroup product of and is . In this example,

References

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  1. ^ O'Connor, John J.; Robertson, Edmund F., "Walter Douglas Munn", MacTutor History of Mathematics Archive, University of St Andrews