# Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$

My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$.

I'm a bit confused and I don't know what is the notation when we work with these kind of constructions but I mean something like:

$\langle H, \sqsubset_1,\sqsubset_2,...,\sqsubset_i \rangle$ but I don't know if is correct because we have infinite partial orders.

or $\{\langle H, \sqsubset_i \rangle:i\in I\}$ we have a set of partial order structures on $H$.

I already know that there are infinite partial orders on every set, but I'm asking if

Do exists a theory that studies sets structured by and infinite set of partial orders as single objects, where are explored requirements and derived theorems?

We have something similar for example when we pass from structures with one binary operation to fields and rings and in general sets equipped with two binary operations.

Update

Using this notation (if I understand well) we can denote this structure with

$\mathfrak H=\langle H, \{\sqsubset_i \}_{i\in I} \rangle$ and $|I|\ge \aleph_0$

UPDATE: To be more specific that is an example of what kind of structures I'm talking about:

For every infinite set $H$ I have a set of relations $\{\sqsubset_a:a\in H\}$ that are strict partial orders on $H$.

with that properties:

$i)$ if $\forall a,b(a,b\in H)(a\neq b)$ than $\sqsubset_a \cap\sqsubset_b=\varnothing$

$ii)$ $\forall a(a\in H)$ and $\forall A \in \mathcal P(H)$ if $a \in A$ then $a$ is the $\sqsubset_a$-minimal element of $A$

To be more clear here an important comment of Kevin Carlson

"Surely such things are used somewhere in the wide world of human endeavor. I don't know where. But the orders must definitely interact in well-defined ways in any field where they're found to be useful, or as we've both observed the structures are too general to be of interest."

Well the question is about an axiomatic study of these kind of structures and example of compatible properties that are already useful in some field of mathematics.

UPDATE: I just discovered about Multi-ordered Posets as sets equipped with a finite number of partial order relations in Bishop, Killpatrick MULTI-ORDERED POSETS

A definition is given and these structures are introduced to generalize the concept of differential posets: this is one kind of structure I am interested in but the topic is too much advanced for me and the goal of the paper is not very interesting too (for me)...but there are not external references to these structures in this paper or to an introduction and is what I am searching for.

• Are there any known relations between the given partial orders on $H$? May 7, 2013 at 16:46
• @ArthurFischer I don't know if is important to say the properties of my particular structure here, becuase I wanted to find the more general results as possible about these structures. but my structure have some properties: May 7, 2013 at 18:16
• @ArthurFischer A) for example I know (I'm still studyng it) that different elements induces different partial orders (aka one is not a sub order of another and it does hold for some elements, but I don't have a proof that all the relations are pairwise disjoint). and B) for every subset $A\in H$ and for all $a\in H$ if $a\in A$ then in $\langle A, \sqsubset_a \rangle$ the smaller element is $a$. May 7, 2013 at 18:17
• @ArthurFischer now that I read again my second comment seems me a bit unclear, but I mean that my structure is equipped form a set of p.orders $\{ \sqsubset_a : a\in H\}$ and if $a\neq b$ then $\sqsubset_a\neq \sqsubset_b$ May 9, 2013 at 7:18
• I'm unclear on what you're asking for. As you say, it's easy to endow any infinite set with infinitely many distinct partial orders, so there's no way to expect any non-trivial results in general. Certainly such structures are perfectly allowable in model theory, though I can't tell whether that's how you're using the word "structure." Your $\mathfrak{H}$ notation is fine. So I can be a tiny bit useful, note your relations are certainly not pairwise disjoint, since partial orders are reflexive. May 10, 2013 at 9:04

I am not sure if this helps but if for each $a \in H$ we define $$\sqsubset_{a} = \{ (a,b) \colon b \in H \text{ and } a \neq b \}$$ then we get a family of strict partial orders that satisfy your conditions. This demonstrates consistency. No claim is made for usefulness of this example.