In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero.

1) Is there a standard terminology for operads with and without $O(0)$?

2) Is there standard terminology for topological operads where $O(0)$ is a point, vs. $O(0)$ not being a point?

Although it's less important I'd be curious if people have examples where these distinctions are interesting.

Since any operad acts on its $O(0)$ part perhaps the $O(0)$ part should be called something like its "base"? But then "baseless operad" would sound kind of pejorative.

reduced. I like that terminology better. – Mike Shulman Jun 27 '10 at 3:12