# Definition of tautological action

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."

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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.

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What don't you like about the term, is it misleading? Is there a better term for it? –  user50229 Dec 7 '12 at 23:10
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. –  Qiaochu Yuan Dec 8 '12 at 0:45
I just found that in Bourbaki's Algebra I, page 25 example 3, they refer to this action as the canonical action. –  user50229 Dec 26 '12 at 17:45