2
$\begingroup$

What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions?

For reference the particular sentence is: "A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being effective, i.e., if two elements of the semigroup have the same action, then they are equal."

1 Answers 1

1

I don't like this terminology. What it appears to mean is the following: you can think of a transformation semigroup either concretely as a collection of functions from a set $S$ to itself closed under composition, or abstractly as an abstract semigroup $G$ (namely the functions above) together with a faithful (effective) action of $G$ on $S$. The tautological action is this action.

  • 0
    What don't you like about the term, is it misleading? Is there a better term for it?2012-12-07
  • 0
    It's just not very descriptive. I don't know a better term. I just wouldn't talk about transformation semigroups in the first place.2012-12-08
  • 1
    I just found that in Bourbaki's Algebra I, page 25 example 3, they refer to this action as the _canonical action_.2012-12-26
  • 0
    The answer to the question of the original post ist still unclear. Can anbody else comment on this?2015-11-18