  We consider effects of inter-species attraction on two-component gap solitons
(GSs) in the binary BEC with intra-species repulsion, trapped in the
one-dimensional optical lattice (OL). Systematic simulations of the coupled
Gross-Pitaevskii equations (GPEs) corroborate an assumption that, because the
effective mass of GSs is negative, the inter-species attraction may
\emph{split} the two-component soliton. Two critical values, $\kappa_{1} $ and
$\kappa_{2}$, of the OL strength ($\kappa $) are identified. Two-species GSs
with fully overlapping wave functions are stable in strong lattices ($\kappa
>\kappa_{1}$). In an intermediate region, $\kappa_{1}>\kappa >\kappa_{2}$, the
soliton splits into a double-humped state with separated components. Finally,
in weak lattices ($\kappa <\kappa_{2}$%), the splitting generates a pair of
freely moving single-species GSs. We present and explain the dependence of
$\kappa_{1}$ and $\kappa_{2}$ on thenumber of atoms (total norm), and on the
relative strength of the competing inter-species attraction and intra-species
repulsion. The splitting of asymmetric solitons, with unequal norms of the two
species, is briefly considered too. It is found and explained that the
splitting threshold grows with the increase of the asymmetry.
