comparing two sets in set theory

Question

I have two sets A and B and this condition holds:

$\forall x \in A , y \in B: x \leq y$

Is there any standard term to describe the relation of A and B? something like $A \leq B$?

Thanks for your help.

Answer 1

You can always define your own operations. However, defining $\leq$ the way you described it is difficult because their is no one unique ordering of the sets $A$ and $B$. Sure, sets like $\mathbb{N}$, $\mathbb{Q}$ etc. have natural orderings but you can reorder them and in general it is not unique. So we combine the sets with an ordering and get a structure.

Let $\mathfrak{A} = (A,\leq_A)$ be a structure where $\leq_A$ is an (linear) ordering. Let $\mathfrak{B} = (B,\leq_B)$ and $\mathfrak{C} = (C,\leq_C)$ be substructures of $(A,\leq)$. This means that $B,C \subseteq A$, $\leq_{A|B} = \leq_B$ and $\leq_{A|C} = \leq_C$.

Then we write $\mathfrak{B} \leq_\mathfrak{A} \mathfrak{C}$ if and only if for each $b \in B$ and each $c \in C$ it holds that $b \leq_A c$.

This notation is a bit bloated but it works I think. Maybe you can simplify it for your purposes. For example fix $A$ or maybe $\leq_A,\leq_B,\leq_C$ are always natural orderings?