the rank of matrix over subset lattice

Question

There is a matrix over subset lattice. Are there any row elements of a matrix that can linearly represent any row? If so, what is the definition of linear representation? Does there exist scalar $a_1,a_2,\ldots,a_k$ such that $y=\vee a_ix_i$, What is the meaning of scalar? Is it also a subset?

Answer 1

Let me try to provide some definitions/context. Suppose $\mathsf{P}(X)$ denotes the powerset of a given set $X$. As is obvious, $(\mathsf{P}(X), \cap, \cup, \emptyset, X)$ is a lattice.

Following a definition due to Maragos, the set of maps $\{1, 2, \cdots, n\} \to \mathsf{P}(X)$ which we call a weighted lattice. Let $V:= \mathsf{P}(X)^n$ and $W:=\mathsf{P}(X)^m$. We call elements of weighted lattice a vector. A matrix $A$ with coeficients in $\mathsf{P}(X)$ acts on a vector $v \in V$ via $$ [A v]_i = \bigcup_{j=1}^n [A]_{ij} \cap v_j. $$ inducing a join-preserving map $A: V \to W$ (something you need to check). Here join is the join in the product i.e. $v \vee v^{'} := (v_1 \cup v_1^{'}, \dots, v_n \cup v_n^{'})$. Because $A$ (as a lattice map) is $\vee$-preserving, it sends $\vee$-irreducibles to $\vee$-irreducibles (much like linear maps send basis elements to basis elements).

$\vee$-irriducibles of $V$ look like tuples such that every coordinate is a join-irreducible of $\mathsf{P}$ i.e. a singleton.

Regarding independence, the natural generalization of linear independence in this setting is: a collection of vectors is independent $\{v_1, v_2, \cdots, v_n\}$ if for $A = \left[v_1 \vert v_2 \vert \cdots \vert v_n \right]$, then $A x = 0$ implies $x = 0$. (Here, $0$ is the vector whose entries are all empty).