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 

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ethan Akin
    • 1
  1. 1.The City CollegeNew YorkUSA

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