Verification: Give a definition of a $G$-Set so that it may be viewed as an algebra

Question

A $G$-Set for those who may not be familiar with the terminology is as follows:

Let $G$ be a group and $S$ be a set. $G$ acts on $S$ define by the map $\star : G \times S \to S $ where $e \star s = s$ and $g\star (h \star s) = gh \star s, \forall s \in S, g,h \in G $ and so $\langle S, \star \rangle$ is a $G$-set

An algebra is a pair $\langle A, F \rangle $ where $A$ is the universe and $F$ is the family of operations on $A$

Now I need to give a definition that presents $G$-sets as an algebra which are equivalent to the above definition of a $G$-set

I thought one such definition would be :

$\langle S, \star \rangle =\langle S, \star_{g},\star_{h} ... \rangle$ (***) where we have $\star = \{\star_{g} : g \in G\}$

$\star_{g} s = g \star s, \forall g \in G, \forall s \in S$

That is for all elements of $G$, each element becomes associated with a unary operation in the signature of (***) .

so we would have $\star_{e}s = e \star s = s, \forall s \in S$ for $\star_{g}(\star_hs) = \star_g(h \star s) = g \star (h \star s)) = hg (\star s) = \star_{gh}s$

This seems valid as we have our universe $S$ with a signature of operations and can axiomatize it by the above equations.

Answer 1

Yes, what you describe is the standard way of describing a $G$-set as an algebra. This is for fixed $G$. You can also give equational axioms for the theory of group actions (where the group is also allowed to vary), but you have to use multi-sorted logic.

Answer 2

You can take a $\Bbb{K}$-vector space $V$ (for some field $\Bbb{K}$), whose dimension is equal to the cardinality of the set $S$. That is, for each $s \in S$, you have a basis element $e_s \in V$. Then your construction is essentially the so-called "group algebra" $\Bbb{K}[G]$, realized as a sub-algebra of the endomorphisms of $V$. So what you call $\star_g$ is realized concretely here as the linear map defined by

$$ \star_g(e_s) = e_{g \star s} $$

So what I'm getting at is that $\mathrm{End}_\Bbb{K}(V)$ is an algebra, and the maps $\star_g$ generate a sub-algebra.

Answer 3

Your definition is fine, although using $\star$ for what is (as you point out) a unary operation maybe isn't ideal.

For a group $G$ acting on a set $X$, define the $G$-set $\langle X, \overline{G}\rangle$ to be the algebra with universe $X$ and operations $\overline{G} = \{\bar{g} \mid g \in G\}$, one (unary) operation for each $g\in G$. Then $\bar{g}x$ would denote the action of the group element $g$ on the element $x\in X$. As you point out, we require $\bar{e} = \operatorname{id}_X$ (the identity function on $X$), and $\overline{gh} = \bar{g} \circ \bar{h}$.

Essentially, this is what you have already figured out for yourself, in slightly different notation.

See also Bill Lampe's very nice and concise presentation of $G$-sets as universal algebras, which develops some of the properties of $G$-sets from this perspective.