Definition 4

Let K be a non-discrete valued division ring and E be a left vector space on K with two subsets A and B. We say that A absorbs B if there exists α > 0 such that λA ⊃ B for every λ ∈ K with |λ| ≥ α (or equivalently if μB ⊂ A for μ ≠ 0 and |μ| ≤ α⁻¹). A set A of E is called absorbent if it absorbs every set consisting of a single point.

Let A be a balanced set of E; for it to absorb a set B of E it is sufficient that there exists λ ≠ 0 such that λA ⊃ B; in fact, for |μ| ≥ |λ|, we have λA = (λμ⁻¹) μA, and as μA is balanced and |λμ⁻¹| ≤ 1, it follows that λA ⊂ μA, and thus B ⊂ μA. In particular for a balanced set A of E to be absorbent, it is necessary and sufficient that for every x ∈ E, there exists λ ≠ 0 in K such that λx ∈ A. Every absorbent set of E generates the vector space E. Every finite intersection of absorbent sets is an absorbent set.