Relaxation of valuation axioms?

Question

Let $K$ be a field, and $v:K\rightarrow \mathbb{R}$ be a valuation, i.e. a function such that, for all $x,y\in K$: $v(x)\geq 0$, $v(x+y)\leq v(x)+v(y)$ (if $v$ is Archimedian, alternatively $v(x+y)\geq \min(v(x),v(y))$) and $v(xy)=v(x)v(y)$. My question is, are there meaningful objects that come out of relaxing any of these conditions? I would be particularly interested to know if one can relax to a "super" or "sub" valuation satisfying $v(xy)<v(x)v(y)$ or $v(xy)>v(x)v(y)$.

Answer 1

Another broad generalisation is that of filtered rings (and modules). Some common filtrations are basically valuations except you allow $v(xy) \ge v(y)+v(y)$.

The theory is rich. Concepts you can search for are the associated graded ring (module, algebra), or the Artin-Rees condition, or microlocalisations. Bourbaki has an entire chapter about filtrations and graduations (chapter III in Algèbre Commutative), another bible is https://link.springer.com/book/10.1007/BFb0067331.

Answer 2

The natural relaxation is to replace $\Bbb{R}$ by any ordered abelian group, for example $$v(\sum_{n,m} c_{n,m}x^ny^m)=\inf_{n,m,c_{n,m}\ne 0} n+m\epsilon, \qquad v(f/g)=v(f)-v(g)$$ on $\Bbb{Q}(x,y)$.

The valuation ring (elements with valuation $\ge 0$) is $$\Bbb{Q}[x,y]_{(x,y)}+x\Bbb{Q}(y)[x]_{(x)}$$