We have used a self-consistent semi-empirical molecular orbital method to investigate whether the adsorption properties of K atoms and formation of the K adsorbate chains or clusters in the low-coverage regime can be influenced by the nature of the semiconductor surface: perfect or stepped. In the process, we are able to determine the microscopic structures of monatomic and diatomic K molecules on perfect and stepped GaAs(110) surfaces. Our results for K adsorption on the perfect GaAs(110) surface are consistent with recent scanning tunnelling microscopy (STM) observations for Na on GaAs(110), with the stable site for K being the bridge site encompassing one Ga and two As surface atoms. The equilibrium geometry for diatomic K has the second K atom occupying the next-nearest-neighbour bridge site, strongly supporting the formation of an open linear structure parallel to the surface atomic zigzag chains. The calculated K-K distance in this equilibrium configuration is 8.02 angstrom, similar to the Na-Na distance (8 angstrom) from the STM experiment. Our results for the stepped GaAs(110) surface suggest that a step is unlikely to assist clustering of K atoms, but, the formation of the linear adsorbate chain appears rather to be influenced by the orientation of the steps. However, the K adsorbates are bound more strongly at the steps than at the bridge sites on the perfect surface.