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."
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.