On disjoint sums in the lattice of linear topologies

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 $\tau_*,\tau_0,\tau_1\in\mathcal L(M)$. We write $\tau_1=\tau_*\sqcup\tau_0$ or say that $\tau_1$ is a disjoint sum of $\tau_*$ and $\tau_0$ if $\tau_1=\inf\{\tau_0,\tau_*\}$ and $\sup\{\tau_0,\tau_*\}$ is the discrete topology. Given $\tau_1,\tau_0\in\mathcal L(M)$, we say that $\tau_0$ is a disjoint summand of $\tau_1$ if $\tau_1=\tau_*\sqcup\tau_0$ for a certain $\tau_*\in\mathcal L(M)$. Some necessary and some sufficient conditions are proved for $\tau_0$ to be a disjoint summand of $\tau_1$.