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$M$-mapping; topological group; Maltsev space; $\aleph_0$-cellularity
We consider $M$-mappings which include continuous mappings of spaces onto topological groups and continuous mappings of topological groups elsewhere. It is proved that if a space $X$ is an image of a product of Lindelöf $\Sigma$-spaces under an $M$-mapping then every regular uncountable cardinal is a weak precaliber for $X$, and hence $ X$ has the Souslin property. An image $X$ of a Lindelöf space under an $M$-mapping satisfies $cel_{\omega}X\le2^{\omega}$. Every $M$-mapping takes a $\Sigma(\aleph_0)$-space to an $\aleph_0$-cellular space. In each of these results, the cellularity of the domain of an $M$-mapping can be arbitrarily large.
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