Two-dimensional NPT Monte Carlo simulations are performed for binary mixtures of Lennard-Jones particles, D and L. The DD-interactions are equal to the LL-interactions. The optimum DL-distance is changed using one parameter in the Lennard-Jones potential by s=σDL/σDD, resulting in nonadditive mixtures. Simulations of the solid phase yields various crystal structures, like triangular, square and hexagonal solid solutions, which are compared with our previous results of analogous three dimensional mixtures [M. J. Vlot et al., J. Chem. Phys. 107, 4345 (1997). In most cases, the 2D crystals are layers from the corresponding 3D compounds. The Gibbs free energy is measured to determine the stability limits. Raising s from s=1 (ideal mixture) to 1.2 a remarkable chainlike order, to which we refer to as spaghetti order, develops continuously in the solid phase. Increasing s further, results in a transition to a liquid, again with clear spaghetti ordering. Topologically these structures show similarities to structures occurring during spinodal decomposition, but our spaghetti solid and liquid are much more stable than the demixed states. We suggest a structure parameter deduced from the partial radial distribution functions (DD and DL), to detect the spaghetti structure.