On Disjoint Sums in the Lattice of Linear Topologies

Abstract

Let M be a vector space over a skew-field equipped with the discrete topology, $\mathcal{L}$(M) be the lattice of all linear topologies on M ordered by inclusion,and τ_*, τ₀, τ₁ ∈ $\mathcal{L}$(M). We write τ₁ = τ_* ⊔ τ₀ or say that τ₁ is a disjoint sum of τ_* and τ₀ if τ₁ = inf{τ₀, τ_*} and sup{τ₀, τ_*} is the discrete topology. Given τ₁, τ₀ ∈ $\mathcal{L}$(M), we say that τ₀ is a disjoint summand of τ₁ if τ₁ = τ_* ⊔ τ₀ for a certain τ_* ∈ $\mathcal{L}$(M). Some necessary and some sufficient conditions are proved for τ₀ to be a disjoint summand of τ₁.