Munn semigroup

Last updated April 26, 2019

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]

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

In mathematics, a join-semilattice is a partially ordered set that has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.

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.

In the mathematical field of ring theory, a principal ideal is an ideal $in a ring that is generated by a single element of through multiplication by every element of . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element, which is to say the set of all elements less than or equal to in .$

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 := $\bigcup _{e,f\in E}$ { T_e,f : (e, f) ∈ U }.

The semigroup's operation is composition of partial mappings. 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.

In mathematics, a partial function from X to Y is a function $f : X' \to Y$ , for some subset X′ of X. It generalizes the concept of a function $f : X \to Y$ by not forcing f to map every element of X to an element of Y. If X′ = X, then f is called a total function for emphasizing that its domain is not a proper subset of X. Partial functions are often used when the exact domain, X, is not known. In real and complex analysis, a partial function is generally called simply a function.

In abstract algebra, the set of all partial bijections on a set X forms an inverse semigroup, called the symmetric inverse semigroup on X. The conventional notation for the symmetric inverse semigroup on a set X is $or In general is not commutative.$

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

Theorem

For every semilattice $E$ , the semilattice of idempotents of $T_{E}$ is isomorphic to E.

Example

Let $E=\{0,1,2,...\}$ . Then $E$ is a semilattice under the usual ordering of the natural numbers ( $0<1<2<...$ ). The principal ideals of $E$ are then $En=\{0,1,2,...,n\}$ 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 $T_{m,n}=\emptyset$ if $m\not =n$ . The semigroup product of $1_{Em}$ and $1_{En}$ is $1_{E\operatorname {min} \{m,n\}}$ . In this example, $T_{E}=\{1_{E0},1_{E1},1_{E2},\ldots \}\cong E.$

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