Set ordered in two different ways.

Question

First of all i must inform you that this is general question, so i would like to know only general information.

Let $A, B$ be some sets and let $C\subseteq A\times B$.

Let $(a,b) , (c,d) \in C$

There are two different orders:

• $(a,b)\le_{1}(c,d) \iff a\subseteq c$
• $(a,b) \le_{2}(c,d) \iff b\subseteq d$

Consider now the structure $P=(C, \le_{1}, \le_{2})$. It is different from ordinary partially ordered set, because of the one extra order

Is there any known theory that tells about the structure above?

I would like to know some PDF or books about this structure, its properties and known results.

Regards

Answer 1

Those relations $\leq_1$ and $\leq_2$ are not partial order relations;
for example, with $b\neq d$, we have $$(a,b) \leq_1 (a,d) \quad\text{ and }\quad (a,d) \leq_1 (a,b).$$ These are quasi-orders (also pre-orders).
I don't know of any book discussing them, or freely available pdf, but here are two articles which include something about them, including that structure with two quasi-orders:

• A. Urquart, A topological representation theory for lattices, Algebra Universalis, 8 (1978) 45-58,
• G. Markowski, The factorization and representation of lattices, Trans. Amer. Math. Soc., 203 (1975) 185-200.