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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.

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It is not clear to me that $\Omega$-algebras ought to act on sets at all. –  Qiaochu Yuan Jul 25 '12 at 0:00
    
@Hooman Could you give more background/context? –  Dylan Moreland Jul 25 '12 at 1:06
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As far as I can see you need additional operations on the collection of endomorphisms of your sets. So perhaps if you passed from Sets to some enriched category? I can't say I can think of any interesting example of such a structure though, and it seems quite asymmetric. I'm wondering if the notion of a group action shouldn't be considered more of a construction on sets than on groups, arising due to the existence of a canonical group structure on the set of automorphisms of a set (or more generally the objects of a category). –  Tilo Wiklund Jul 25 '12 at 1:29
    
@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. –  Hooman Jul 25 '12 at 7:26

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