Re: a simple question on algebra (left/right identities and left/right inverses).

In article <dgu9bs$2u1d$1@xxxxxxxxxxxxxxxxxx>,
Arturo Magidin <magidin@xxxxxxxxxxxxxxxxx> wrote:

>His actual question was:
>
> Suppose S is a semigroup for which the following two are true:
> (i) There exists e in S such that for all a, ea=a.
> (ii) There does NOT exist an e' in S such that for all a, ae' = a.
>
> Does it FOLLOW that for all a in S, there exists a' such that
> aa'=e?
>
> I.e., can you have a semigroup with a left identity, no right
> identity, and no right inverses, or does the existence of a
> left-but-not-right identity imply the existence of right inverses?

If that's the question, then an example is G^2 (for any group G with
more than one element) with the operation
(x,y) * (z,w) = (xz, xw).
Left identities are [1,y] for every y.
No right identity, and no right inverse, since there's no way
to recover y from (x,y) * anything.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.