Copies of the sequence space $\omega$ in $F$-lattices with applications to Musielak−Orlicz spaces

Let $E$ be a fixed real function $F$-space, i.e., $E$ is an order ideal in $L_0(S,\mathrm{\Sigma },\mu )$ endowed with a monotone $F$-norm $\Vert \Vert$ under which $E$ is topologically complete. We prove that $E$ contains an isomorphic (topological) copy of $\omega$, the space of all sequences, if and only if $E$ contains a lattice-topological copy $W$ of $\omega$. If $E$ is additionally discrete, we obtain a much stronger result: $W$ can be a projection band; in particular, $E$ contains a~complemented copy of $\omega$. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of $\omega$ by a Musielak−Orlicz space is characterized as follows. (1) A sequence space ${\ell }_{\mathrm{\Phi }}$, where $\mathrm{\Phi }=({\phi }_n)$, contains a copy of $\omega$ iff $\underset{n\in \mathbb{N}}{inf}{\phi }_n(\mathrm{\infty })=0$, where ${\phi }_n(\mathrm{\infty })=\underset{t\to \mathrm{\infty }}{lim}{\phi }_n(t)$. (2) If the measure $\mu$ is atomless, then $\omega$ embeds isomorphically into $L_{\mathcal{M}}(\mu )$ iff the function ${\mathcal{M}}_{\mathrm{\infty }}$ is positive and bounded on some set $A\in \mathrm{\Sigma }$ of positive and finite measure, where ${\mathcal{M}}_{\mathrm{\infty }}(s)=\underset{n\to \mathrm{\infty }}{lim}\mathcal{M}(n,s)$, $s\in S$. In particular, (1)' ${\ell }_{\phi }$ does not contain any copy of $\omega$, and (2)' $L_{\phi }(\mu )$, with $\mu$ atomless, contains a~copy $W$ of $\omega$ iff $\phi$ is bounded, and every such copy $W$ is uncomplemented in $L_{\phi }(\mu )$.