  Let $G$ and $H$ be locally compact groups with fixed two-side-invariant Haar
measures. A polyhomomorphism $G\to H$ is a closed subgroup $R\subset G\times H$
with a fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the
Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a
multi-valued map sending points to sets equipped with 'uniform' measures. For
polyhomomorphsisms $G\to H$, $H\to K$ there is a well-defined product $G\to K$.
The set of polyhomomorphisms $G\to H$ is a metrizable compact space with
respect to the Chabauty--Bourbaki topology and the product is separately
continuous. A polyhomomorphism $G\to H$ determines a canonical operator
$L^2(H)\to L^2(G)$, which is a partial isometry up to scalar factor. As an
example, we consider locally compact infinite-dimensional linear spaces over
finite fields and examine closures of groups of linear operators in semigroups
of polyendomorphisms.
