This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^\*$ be its Grothendieck group. That is, $T^\* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+e$ for some $e \in T$. 


Let $i : T \to T^\*$ be the natural inclusion map, where $i(t) = [(t,0)]$, and let $K$ denote the kernel of this map. 


Does the kernel $K \subseteq T$ have a common name in the literature? What are elements $k \in K$ called?



Kernel elements for the Grothendieck group map of a commutative monoid
I don't know a name for $K$ either, but here is a suggestion.


The correct definition of $\sim$ is $(a,b) \sim (c,d) \Leftrightarrow \exists e \in T ~ (a+d+e=b+c+e)$. In particular $(a,b) \sim (0,0) \Leftrightarrow \exists e \in T : a+e=b+e$. I would suggest to call such elements $a,b$ *stably equal*. Remark that this coincides with the terminology *stably isomorphic* from topological $K$-theory (where $T$ is the commutative monoid of $\cong$-classes of vector bundles on a space $X$ and one usually takes w.l.o.g. $e$ to be the class of a trivial vector bundle when $X$ is paracompact). In particular, $a \in K \Leftrightarrow \exists e \in T : a+e=e$, in which case one may say that $a$ is *stably zero*.


