What is the formal name of this set-related concept?

Question

I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name?

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Let $S$ be a family of finite sets.

Say that $S$ is 1-dimensional if there exists a sequence of sets $W_j$, for $j=1,2,\dots$, such that for every $j$:

• $|W_j|=j$, and
• For every set $s\in S$, either $s\subseteq W_j$ or $W_j\subseteq s$.

For example:

• The family of "rays" in $\mathbb{N}$, { {1}; {1,2}; {1,2,3}; ... }, is 1-dimensional. For every $j$, take $W_j=\{1,\dots,j\}$.
• The family of "squares" in $\mathbb{N}\times \mathbb{N}$, { {(1,1)}; {(1,1),(1,2),(2,1),(2,2)}; {(1,1)...(1,3),(2,1)...(2,3),(3,1)...(3,3)}; ... }, is 1-dimensional. For every $j$, take $W_j$ as a subset of the "square" of side-length $\lceil \sqrt j\rceil $ that contains the "square" of side-length $\lfloor \sqrt j\rfloor$.
• The family { {1}; {2}; {3}; ... } is not 1-dimensional. No choice for $W_1$, for example, satisfies the definition.

For every integer $d\geq 1$, say that $S$ is $d$-dimensional if there exists a collection of sets $W_{j,t}$, for $j=1,2,\dots$ such that for every $j$:

• $|W_{j,t}|=j$ for all $t$, and
• The index $t$ ranges between 1 and $j^{d-1}$ (i.e, there are $j^{d-1}$ subsets for every $j$), and
• For every set $s\in S$, there exists some $t \in 1,\dots, j^{d-1}$ such that, either $s\subseteq W_{j,t}$ or $W_{j,t}\subseteq s$.

For example:

• The family of "rectangles" in $\mathbb{N}\times \mathbb{N}$, { {(1,1)}; {(1,1),(1,2)}; {(1,1),(2,1)}; {(1,1),(1,2),(2,1),(2,2)}; ... }, is two-dimensional. For every $j$ and for every $t\in\{1,\dots,j\}$, Take $W_{j,t}$ as a subset of the "rectangle" of length $t$ and height $\lceil j/t \rceil$, which contains the "rectangle" of length $t$ and height $\lfloor j/t \rfloor$.
• Similarly, the family of "boxes" in $\mathbb{N}^d$ is probably $d$-dimensional.

I am mainly interested in questions such as: how to characterize / identify $d$-dimensional families; what other interesting $d$-dimensional families exist?

EDIT: The motivation to this definition comes from probability. Let $X$ be a random variable whose possible values are sets from the family $S$. I would like to prove claims such as "If the size of $X$ is at least $j$, then it must contain one of the (deterministic) subsets $W_{j,t}$".

EDIT 2: I now found out that what I called "1-dimensional family" is actually called: Filtration.

So, my question becomes: is there a natural extension of filtration to two or more dimensions (as explained above)?

Answer 1

The notion that you are describing is somewhat related to the notion of the order dimension of a poset, a notion that has been studied in many papers.

The order dimension of a poset $(X,\leq)$ is the smallest (in many cases) cardinal $\mu$ such that there exists a set $I$ with $|I|=\mu$ and a family of linear orders $(\preceq_{i})_{i\in I}$ such that $\leq$ is equal to $\bigcap_{i\in I}\preceq_{i}$.

Equivalently, the order dimension of a poset $(X,\leq)$ is the smallest cardinal $\mu$ such that there is some set $I$ with $|I|=\mu$ and linear orders $(X_{i})_{i\in I}$ such that $(X,\leq)$ is isomorphic to a subposet of $\prod_{i\in I}(X_{i},\leq)$ with the pointwise ordering.