In this paper, we propose a generalized framework for fitting Boolean matrix factorization models to binary data. In this generalized setting, the binary rank-1 components of the underlying model can be combined by any Boolean function, thus extending the standard Boolean matrix factorization model, where the combination is restricted to the logical ‘OR’ function. We introduce two algorithms relying on a relaxation of the binary constraints on the factors of the model and on a polynomial representation of the Boolean function that combines the rank-1 components. One of the algorithms is based on the gradient descent optimization method, while the other is based on block coordinate descent. A detailed presentation of the algorithms is given, along with numerical experiments both on simulated and real datasets. A comparison with other algorithms from the literature is presented in the standard Boolean matrix setting allowing to assess the advantages and shortcomes of the proposed methods in terms of factor retrieval and data denoising performance, convergence behavior and time complexity.