The theory of attribute implications is a fundamental line of research in formal concept analysis. The fully true attribute implications are the most significant attributes in the Boolean framework, and also play a fundamental role in the fuzzy one. This paper shows that the residuated concept lattice framework satisfies natural extensions of the equivalences among the definition of validity and the characterizations using the extents of the subsets of attributes. Moreover, it introduces sufficient conditions in order to obtain analogous equivalences in the multi-adjoint framework.