Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
2  
I assume by the kernel you mean the inverse image of 0. It is not usual to use the term kernel for the inverse image of the identity in semigroup theory since in general it is not interesting. The fibers of a semigroup homomorphism can all have radically different cardinality. I know of no name for $K$ in semigroup theory although perhaps in K-theory or some place else it has a name. –  Benjamin Steinberg Jan 26 '13 at 3:27
2  
Here's a negative result: no name given in Ogus's log geometry book. –  S. Carnahan Jan 26 '13 at 10:35

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
    
Thanks for the correction & suggestion, Martin! –  Tom LaGatta Jan 27 '13 at 4:18
    
Similarily one could say that for a function $f : X \longrightarrow X$ two elements $x,y \in X$ are stably equal if $f^n(x)=f^n(y)$ for some $n \in \mathbb{N}$ (which means that they are equal in the localization $X_f$). –  Martin Brandenburg Jan 27 '13 at 10:41

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.