# Ellis Semigroups and Ellis Actions

• Ethan Akin
Chapter
Part of the The University Series in Mathematics book series (USMA)

## Abstract

A semigroup S is a nonempty set with an associative (usually not commutative) multiplication map M: S ×SS. For p, qS, 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 × XX where S is a semigroup, for pS 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, qS:
$${\Phi ^p} \circ {\Phi ^q} = {\Phi ^{pq}}$$
(6.4)
Thus M defines an action of S on itself.

## Keywords

Ellis Action Compact Space Semi Group Follow Diagram Commute Transitive Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.