Munn semigroup

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In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Douglas Munn (1929–2008).[1]

Construction's steps[edit]

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 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[edit]

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

Example[edit]

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 . In this example,

References[edit]