Bases of closure systems over residuated lattices

• The problem of bases of closure systems arising in data with grades is described.
• Decomposition of closure operators into two simpler ones is found.
• Methods for computing bases are provided.

We present results on bases of closure systems over residuated lattices, which appear in applications of fuzzy logic. Unlike the Boolean case, the situation is not straightforward as there are two non-commuting generating operations involved. We present a decomposition theorem for a general closure operator and utilize it for computing generators and bases of the closure system. We show that bases are not unique and may in general have different sizes, and obtain a constructive description of the size of a largest base. We prove that if the underlying residuated lattice is a chain, all bases have the same size.