The associative operad with more elements still associative?

Question

So the associative operad is defined as $P(n) = a_n$ for each $n$. What happens if we set $P(3) = a_3$ but add more elements to the other $P(n)$ for example if $P(5) = \{x,y,z\} $. Will an algebra over this still be an associative algebra?

Answer 1

As long as you only have one element in $P(2)$ and $P(3)$, any algebra over this operad will have an associative binary product (the product is the realization of the element in $P(2)$, and the fact that there is only one element in $P(3)$ shows that this product is associative).

If you add extra operations in higher arity, then the algebras will also have other operations, which may or may not be related to the binary product depending on the relations you put in your operad.

Note that of course once you add those operations, you have to put in your operad all the ways they combine with each other.