Nice 0-ification of monoids? 

When dealing with a non-unital algebra A over a field k, there are two ways of adding a 1 to it:

  • The unitalization A\times k with product (a,\lambda)(b,\mu):=(ab+\lambda b +\mu a, \lambda\mu)
  • The multiplier algebra M(A).

Each of these constructions has its meaning and its place. A while ago I already talked at the ARTS blog on how these constructions relate to different compactifications.

Now, let M be a (multiplicative) monoid (or a monoid object in a monoidal category, if you are that kind of person). Assume that I want to add a 0 to that monoid (i.e. an element such that a\cdot 0 = 0 for all a \in M). There is an obvious way to do it: just add a formal element 0 to M and define the multiplication as above.

That construction resembles the trivial unitalization of an algebra, and has a similar problem: it is completely oblivious on whether the original monoid already had a 0 element or not (such as the trivial unitalization doesn’t check if the algebra already had a 1).

My question is, is there any construction similar to the multiplier algebra one that adds a 0 (and possibly some other elements) in such a way that if our starting monoid already had a 0, the resulting object is isomorphic to the original one?

Does “adding a 0″ have any geometrical meaning, maybe in the context of toric varieties (which are locally given by the spectrum of monoid rings)?