Munn semigroup

 ( Semigroup Theory ) 

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).

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Construction's steps Let $E$ 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 T _{e, f} as the set of isomorphisms of Ee onto  Ef.

3) The Munn semigroup of the semilattice E is defined as: T _E := $\backslash bigcup\_\{e,f\backslash in\; E\}$ {  T _{e, f} : ( e,  f) ∈ U }.

The semigroup's operation is composition of partial function. In fact, we can observe that T _E ⊆  I _E where I _E is the symmetric inverse semigroup because all isomorphisms are partial one-one maps from subsets of E onto subsets of  E.

The of the Munn semigroup are the identity maps 1 _Ee.

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Theorem For every semilattice $E$, the semilattice of idempotents of $T\_E$ is isomorphic to E.

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Example Let $E=\backslash \{0,1,2,...\backslash \}$. Then $E$ is a semilattice under the usual ordering of the natural numbers (). The principal ideals of $E$ are then $En=\backslash \{0,1,2,...,n\backslash \}$ for all $n$. So, the principal ideals $Em$ and $En$ are isomorphic if and only if $m=n$.

Thus $T\_\{n,n\}$ = {$1\_\{En\}$} where $1\_\{En\}$ is the identity map from En to itself, and $T\_\{m,n\}=\backslash emptyset$ if $m\backslash not=n$. The semigroup product of $1\_\{Em\}$ and $1\_\{En\}$ is $1\_\{E\backslash operatorname\{min\}\; \backslash \{m,\; n\backslash \}\}$. In this example, $T\_E\; =\; \backslash \{1\_\{E0\},\; 1\_\{E1\},\; 1\_\{E2\},\; \backslash ldots\; \backslash \}\; \backslash cong\; E.$

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