Sub-Reduct

The sub-reduct of an ADF $D = (A, L, C)$ under a three-valued interpretation $v$ is $D^v = (A^v, L^v, C^v)$ with $A^v = v^{-1}(\mathsf{t}) \cup v^{-1}(\mathsf{u})$ and $L^v = L \cap (A^v \times A^v)$, where each $C^v_a$ is $C_a$ with every $b$ such that $v(b) = \mathsf{f}$ substituted by $\bot$.