  Let $(x_\alpha)$ be a net in a vector lattice normed by locally solid lattice
$(X,p,E_\tau)$. We say that $(x_\alpha)$ is unbounded $p_\tau$-convergent to
$x\in X$ if $p(\lvert x_\alpha-x\rvert\wedge u)\xrightarrow{\tau} 0$ for every
$u\in X_+$. This convergence has been studied recently for lattice-normed
vector lattices as the $up$-convergence in \cite{AGG,AEEM,AEEM2}, the
$uo$-convergence in \cite{GTX}, and, as the $un$-convergence in
\cite{DOT,GX,GTX,KMT,Tr2}. In this paper, we study the general properties of
the unbounded $p_\tau$-convergence.
