Unbounded $p$-convergence in Lattice-Normed Vector Lattices

A net $x_{\alpha }$ in a lattice-normed vector lattice $(X,p,E)$ is unbounded $p$-convergent to $x\in X$ if $p(\mathrm{\mid }{x}_{\alpha }-x\mathrm{\mid }\wedge u)\stackrel{o}{\to }0$ for every $u\in X_{+}$. This convergence has been investigated recently for $(X,p,E)=(X,\mid \cdot \mid ,X)$ under the name of $uo$-convergence, for $(X,p,E)=(X,\parallel \cdot \parallel ,\mathbb{R})$ under the name of $un$-convergence, and also for $(X,p,{\mathbb{R}}^{X^{\ast }})$, where $p(x)[f]:=\mathrm{\mid }f\mathrm{\mid }(\mathrm{\mid }x\mathrm{\mid })$, under the name $uaw$-convergence. In this paper we study general properties of the unbounded $p$-convergence.