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Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of these tensors. Some examples from the geometry of a pair of complementary distributions are discussed and for this case the above structural and virtual tensors are expressed indoi:10.2478/v10157-012-0026-7 fatcat:ou5qdd5uyreq5ojru2ksrzks3q