  Yang-Baxter bialgebras, as previously introduced by the authors, are shown to
arise from a double crossproduct construction applied to the bialgebra R T T =
T T R, E T = T E R, \Delta(T) = T \hat{\otimes} T, \Delta(E) = E \hat{\otimes}
T + 1 \hat{\otimes} E and its skew dual, with R being a numerical matrix
solution of the Yang-Baxter equation. It is further shown that a set of
relations generalizing q-Serre ones in the Drinfeld-Jimbo algebras U_q(g) can
be naturally imposed on Yang-Baxter algebras from the requirement of
non-degeneracy of the pairing.
