Munn semigroup

In mathematics, the Munn semigroup is built from an arbitrary semilattice E.

Construction's steps

1) For all e in E, we define Ee: = {i ∈ E : i ≤ e} which is an principal ideal of E.

2)We define U the uniform relation on E. U: = { (e, f) ∈ E × E : Ee ≃ Ef}

3) For all (e, f) in U, we define T_e,f as the set of isomorphisms of Ee onto Ef.

4) Finally, the Munn semigroup of the semilattice E is defined as: T_e := ∐ { T_e,f : (e, f) ∈ U }.

The semigroup's operation is the composition of maps. 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 idempotents of the Munn semigroup are the identity maps 1_Ee.

Theorem

For every semilattice E, the idemtpotents' semilattice of T_E is isomorph to E.

Example

Let E={0,1,2,...} a semilattice (0 < 1 < 2 < ...) Then for all n, En={0,1,2,...,n}

So, E_m≃E_n ⇔ m=n

Hence, U={(n,n) : n ∈ E} = 1_E the identity map from E in E. In this case, we say that E is anti-uniform.

Finally, T_n,n = {1_En} where 1_En is the identity map between En and itself.

$\implies T_E = \{1_E0, 1_E1, 1_E2, \ldots \} \cong E. \,$

Here we can remark that as the theorem stipulated, the semilattice of the idempotents of T_E is isomorph to E. In fact, in this example, the whole semigroup is isomorph to E because it is only filled with idempotents.

References

• Howie, John M. (1995), Introduction to semigroup theory, Oxford: Oxford science publication .