It is almost always assumed that the algebraic structure underlying non-binary Low-Density Parity-Check (LDPC) codes are Finite Fields. However, when considering non-binary LDPC belief-propagation (BP) decoding, Finite Fields are actually over constrained. In this contribution, we discuss the minimal requirements of the algebraic structure used for non-binary LDPC decoding which we denote Finite Division Near Ring over a Subtractive Near Group. To verify the requirements, a general Min-Max decoding algorithm is derived that incorporates any algebraic structure fulfilling this minimal requirement set. It is shown that by relaxing the mathematical constraints, the decoding performance of non-binary LDPC codes can be incrementally improved compared to a Finite-Field-based LDPC code without any additional hardware cost.