# Decidability of a relation on Functional space!

Suppose I have this functional space $$(D=\{ a\searrow b; a \in A , b \in B\}, \leqslant)$$ (partial order relation on step functions!),also suppose that relation $$\leqslant_1$$ is decidable on $$(A,\leqslant_1)$$ and relation $$\leqslant_2$$ is decidable on$$(B,\leqslant_2)$$.

My question is that, is there any theorem or lemma which can make any relation between decidability of relation on D with relations on A and\or B?!

D is countable, same as for A and B. We say $$a\searrow b \leqslant c\searrow d$$ if and only if $$\forall x \in \{ countable set\}; a \leqslant_1 x \rightarrow c \leqslant_1 x$$ implies $$b \leqslant_2 d$$