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

Competition between attractive and repulsive interactions in two-component Bose-Einstein condensates trapped in an optical lattice