Ellis Semigroups and Ellis Actions

Abstract

A semigroup S is a nonempty set with an associative (usually not commutative) multiplication map M: S ×S → S. For p, q ∈ S, we write

$$pq = M\left( {p,q} \right) = {M^p}\left( q \right) = {M_q}\left( p \right)$$

(6.1)

In terms of the translation maps, the associative law says

$${M^p} \circ {M^q} = {M^{pq}}{M_p} \circ {M_q} = {M_{pq}}$$

(6.2)

In general, for a function Φ: S × X → X where S is a semigroup, for p ∈ S and x ∈ X, we write

$$px = \Phi \left( {p,x} \right) = {\Phi ^p}\left( x \right) = {\Phi _x}\left( p \right)$$

(6.3)

The map Φ is called a semigroup action when for all p, q ∈ S:

$${\Phi ^p} \circ {\Phi ^q} = {\Phi ^{pq}}$$

(6.4)

Thus M defines an action of S on itself.