Let \(A= (A,F)\) be an algebra and \(K\) be a sublattice of subsets of \(A\times A\). The paper contains generalizations of concepts, known for congruences, like the Compatible Majority Function Property (CM), Chinese Remainder Condition (CRC) and Compatible Function Extension Property (CFE) for functions compatible with \(K\). The authors study mutual interrelations between pairs of these properties, in particular under certain conditions valid in \(K\). Main results: For any \(K\), \(\text{CFE} (K)\Rightarrow \text{CM} (K)\). If \(K\) contains only tolerances, then \(\text{CFE} (K)\Rightarrow S\circ T= T\circ S\) for all \(S,T\in K\). If \(K\) is a complete lattice which is closed under relational product and contains only diagonal relations, then the conditions \(\text{CFE}(K)\), \(\text{CM}(K)\), and \(\text{CRC} (K)\) are equivalent. The paper contains an interesting example showing which implications are not valid in a general case.