Is there a name for an ordered algebraic structure that models the extended integers (i.e. enhanced with infinity)?

Is there a name for an ordered algebraic structure that abstracts the non-negative extended integers (extended in the sense that they include $\infty$), i.e. is there a name for a partially-ordered, abelian monoid with the additional requirement that every $x$, with the possible exception of the maximum, has an inverse? To put it differently, is there a name for a partially-ordered abelian monoid, which, excluding the maximum element (if any), forms a group?

• Just to be sure, do you require the partial order be compatible with the addition? That is, $x \leqslant y$ implies $x+z \leqslant y+z$? Commented Jun 4, 2018 at 11:20
• @J.-E.Pin: Yes. Commented Jun 4, 2018 at 11:30
• How do you define $\infty+(-\infty)$ in your monoid? Commented Jun 4, 2018 at 11:41
• George, I think, is the standard name for them. Named after Prince Regent George IV. Commented Jun 4, 2018 at 11:42
• @Wojowu: Actually, I had in mind a positive partial order (for which $0$ is the minimum). In other words, I am interested in an abstraction of the extended segment $[0,\infty]$ or alternatively $(-\infty,\infty]$, in which case $x+\infty = \infty$. I've now revised my question accordingly. Commented Jun 4, 2018 at 12:03

1 Answer

It seems you are looking for partially ordered groups with a top element. Also known as po-groups --c.f. po-sets.

These are usually defined as sets having an order and a group operation such that the operation is monotone; i.e., order-preserving.

There are also po-monoids and it seems your particular interest is the case of commutative po-monoids with a top element.

• Thanks. However, I'm not sure this is exactly what I'm looking for, since in a partially ordered group with a top element $t$ it can happen that $t+x < t$ for some element $x$, which is not a desirable behavior, since I need $t+x=t$ for every $x$. Commented Jun 27, 2018 at 4:07
• So you're asking for po-groups with an additively absorptive top element ;-) Commented Jul 4, 2018 at 12:46
• This sounds better. Commented Jul 5, 2018 at 7:25
• po-monoids then - a group cannot have an absorptive element Commented Jul 31, 2018 at 19:43