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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
The answer to the question of the original post ist still unclear. Can anbody else comment on this? – Dan Nov 18 '15 at 12:52

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