Let $ \Omega = \left(n_i\right)_{i \in \mathbb{N}}$ be a collection of nonnegative integers. An $\Omega$-algebra on $X$ is a pair $\left(X, \left(w_i\right)_{i \in\mathbb{N}}\right)$ where for all $i \in I$; $ w_i:X^{n_i}\longrightarrow X $ is a $n_i$ ary operation. Now, my question is how one defines the action of an $\Omega$-algebra on a given set, along the line of how we define the action of a group on a set.
How to define an action of an algebra on a set?
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abstract-algebra
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0@DylanMoreland I was trying to appreciate the concept of Operads at Wiki. I came upon this "Algebras are to operads as group representations are to group" and somehow I found my self asking the above question. – 2012-07-25