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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 $b \leqslant_2 d$

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