On pentagon, ten-term, and tetrahedron relations

It is shown that the tetrahedron equation under the substitution R₁₂₃ = S̄₁₃P₂₃S₁₃, where P₂₃ is the permutation operator, is reduced to a pair of pentagon and one ten-term equations on operators S and S̄. Examples of infinite dimensional solutions are found. O-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra, provide a particular algebraic solution to the problem.