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.