My topology teacher appears to call a monoid a set with an associative binary operation, but with no identity element. According to wiki, this is the definition of a semi-group, although they remark that some authors define semi-groups as having an identity (i.e. synonymously to monoid). But they don't say on the monoid article that some authors take monoid to mean an associative magma(groupoid) with no identity. So, does my teacher simply has the definitions mixed up or do some authors effectively call 'monoid' an associative magma(groupoid)? I wanted to ask here before throwing the "Sir professor, according to wikipedia, you're wrong" at him. I'm sure that's understandable.
IMOH, arguments over terminology are politically dangerous, only attempt this if you are on good terms with your professor, or want to start a fight. In my experience, semi-group means associative binary operation, and monoid means associative binary operation with identity element. Wolfram's mathworld agrees: http://mathworld.wolfram.com/Monoid.html, I usually trust Wolfram for ORTHODOX definitions, wikipedia is good at bringing in side issues and lesser known usage (ok, that's my subjective opinion).
Yeah, I wouldn't recommend telling him he's "wrong", but you might want to tell him that you've seen it mean something else in (many) other standard references. Like, for example, in Rotman's Theory of Groups.
You could tell him. Though he's likely to be unimpressed that you used Wiki as a source, and even less impressed that you're worrying about this than actually learning the course. It's just a name, and mostly unimportant. It is the definition that is important. Ok, it might cause you some confusion when looking in other textbooks.
Quite a lot of even modern mathematical definitions are not standardised with subtle and not so subtle differences cropping up all over the place. Many books will even use different "derivation trees" if you will to arrive at some concepts and objects earlier or later than others would.