Boolean matrix factorization (BMF), or decomposition, received a considerable attention in data mining research. In this paper, we argue that research should extend beyond the Boolean case toward more general type of data such as ordinal data. Technically, such extension amounts to replacement of the two-element Boolean algebra utilized in BMF by more general structures, which brings non-trivial challenges. We first present the problem formulation, survey the existing literature, and provide an illustrative example. Second, we present new theorems regarding decompositions of matrices with ordinal data. Third, we propose a new algorithm based on these results along with an experimental evaluation.