  We investigate the phase diagram of antiferromagnetic spin ladders with
additional exchange interactions on diagonal bonds by variational and numerical
methods. These generalized spin ladders interpolate smoothly between the
$S=1/2$ chain with competing nn and nnn interactions, the $S=1/2$ chain with
alternating exchange and the antiferromagnetic $S=1$ chain. The Majumdar-Ghosh
ground states are formulated as matrix product states and are shown to exhibit
the same type of hidden order as the af $S=1$ chain. Generalized matrix product
states are used for a variational calculation of the ground state energy and
the spin and string correlation functions. Numerical (Lanczos) calculations of
the energies of the ground state and of the low-lying excited states are
performed, and compare reasonably with the variational approach. Our results
support the hypothesis that the dimer and Majumdar-Ghosh points are in the same
phase as the af $S=1$ chain.
