On Mon, 19 Apr 2004, Pablo Diaz-Gutierrez wrote:\oplus is frequently read as "direct sum", but in the context quoted above I think Zimmerman is using it to mean cartesian product. (Why he's not using the more common \times I have no idea.)
I think, as with any other symbol, the meaning depends on the context. As of right now, I am doing my homework on differential topology, and the circled product represents the tensor product. I remember some other cases where circled operators were used to denote vector/matrix operations, as opposed to their scalar counterparts.
Pablo,
Thanks to you and the others who directly replied. I now understand that these symbols derive from Lewis Carroll's, "Through the Looking Glass", where things are what you're told they mean. Nothing less and nothing more. My problem, as one respondent pointed out should I chose to use them, is defining just what _I_ mean by their use. What prompted my question is that in the contexts in which I find them, the author does _not_ explain their use. At least, not so a mere ecologist like me can intuit what he means.
Here's an example from Hans Zimmerman's "Fuzzy Sets, Decision Making and Expert Systems", in the section on fuzzy game theory:
"We will start with considering two-person games and specify what is meant by a classical two-person-nonzero-sum game. Let s_ik \in k=1,2 be the ith pure strategy of player k. For any pair {s_i1,s_j2} from S_1 \oplus S_2 there exists a unique real number g_k(s_i1,S_j2) \in G_k which is called the gain of player k."
I've seen these in may other writings of fuzzy sets, logic and system models. Over the years I've learned a lot of mathematics on my own, but it helps to have a dictionary of symbolic usage to which I can refer.
Many thanks, all of you!
Rich
-- Paul
