Results on Boolean matrices Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their many applications to computer science.
I'm interested in understanding what is known about matrices with entries in the Boolean algebra of two elements $\mathbb{B}_2$, endowed with the operation of matrix multiplication
$$(AB)_{ik} = \bigvee_{j=1}^n (A_{ij} \land B_{jk}) ,$$
where $\lor$ and $\land$ are the logical "or" and "and" on $\mathbb{B}_2$ (here $A$ and $B$ are $m \times n$ and $n \times q$ matrices over $\mathbb{B}_2$), and perhaps the operations of matrix-or and matrix-and defined entry-wise.
I wonder how much theory have been developed in analogy with linear algebra and what results are known (are there concept like rank, determinant, nullspace... in this context?)
I haven't found much, except for a 1982 text of Kim "Boolean Matrix Theory and Applications," on which unfortunately I do not have access. Much of the difficulty in searching on this topic is that I keep finding results regarding matrix over $\mathbb{F}_2$.
 A: We can try to produce some of the basic theory of Boolean matrices here to see what makes sense and what does not. For this post, we do not lose much by generalizing to   the Boolean algebras of the form $P(X)$ where $X$ is a possibly infinite set. Everything in this post is well-known and covered in texts on lattice theory, Galois connections, and formal context analysis.
Matrix-linear transformation duality
Suppose that $R\subseteq X\times Y$. Then we can define a function $R^*:P(X)\rightarrow P(Y)$ that preserves arbitrary joins by letting
$R^*(U)=R[U]$ for each $U\subseteq X$. Similarly, if $\phi:P(X)\rightarrow P(Y)$ is a function that preserves arbitrary joins, then we can define a relation $\phi^*\subseteq X\times Y$ where we set $(x,y)\in\phi^*$ if and only if
$y\in\phi(\{x\})$. It is not too hard to show that if $R\subseteq X\times Y$, then $R=R^{**}$ and if $\phi:P(X)\rightarrow P(Y)$ preserves arbitrary joins, then $\phi=\phi^{**}$. We therefore have a one-to-one correspondence between relations (which are like matrices) and maps preserving arbitrary joins (which are like linear transformations). Furthermore, if $R\subseteq X\times Y,S\subseteq Y\times Z$, then $(R\circ S)^*=R^*\circ S^*$, so this correspondence is an isomorphism between the category of all relations and the category of mappings from powersets that preserve arbitrary joins. If $\phi_i:P(X)\rightarrow P(Y)$ preserves arbitrary joins for $i\in I$, then the mapping $\bigvee_{i\in I}\phi_i:P(X)\rightarrow P(Y)$ defined by letting $(\bigvee_{i\in I}\phi_i)(U)=\bigcup_{i\in I}\phi_i(U)$ also preserves arbitrary joins, and $(\bigcup_{i\in I}R_i)^*=\bigvee_{i\in I}(R_i^*)$ whenever $R_i\subseteq X\times Y$ for $i\in I$. Therefore, this functor also preserves arbitrary joins.
Transpose
If $R$ is a relation, then the relation $R^{-1}=\{(y,x)\mid (x,y)\in R\}$ is analogous to the transpose of a matrix.
Nullspace
The notion of the null-space of a matrix generalizes to congruences, closure operators, and closure systems. If $R\subseteq X\times Y$, then the null-space of $R$ is the equivalence relation $\{(U,V)\mid U,V\subseteq X,R[U]=R[V]\}$
If $P$ is a poset, then a mapping $C:P\rightarrow P$ is said to be a closure operator if $x\leq C(x)=C(C(x))\leq C(y)$ whenever $x,y\in P,x\leq y$. We say that a subset $C\subseteq P$ is a closure system if for each $x\in P$, there is a least $y\in C$ with $x\leq y$. If $C$ is a closure operator on a poset $P$, then define a closure system $C^*=\{x\in P\mid C(x)=x\}=C[P]$. If $C$ is a closure system on a poset $P$, then define a closure operator $C^*$ by letting $C^*(x)$ be the least element in $C$ with $x\leq C^*(x)$. If $C$ is either a closure operator or a closure system on a poset $P$, then $C=C^{**}$. We observe that if $C$ is a closure system on a complete lattice, then $C$ is closed under arbitrary meets, so $C$ itself is a complete lattice with a different join operation where if $x_i\in C$ for $i\in I$, then $\bigvee_{i\in I}^{C}x_i=C^*(\bigvee_{i\in I}^Px_i).$ Furthermore, the mapping $C^*:P\rightarrow C$ preserves arbitrary joins in the sense that
$C^*(\bigvee_{i\in I}^Px_i)=\bigvee_{i\in I}^CC^*(x_i)$ whenever $x_i\in P$ for $i\in I$.
If $L$ is a complete lattice, then we say that an equivalence relation $\simeq$ on $L$ preserves arbitrary joins if $x_i\simeq y_i$ for $i\in I$, then $\bigvee_{i\in I}x_i\simeq\bigvee_{i\in I}y_i$.
If $E$ is an equivalence relation on a complete lattice $L$ that preserves arbitrary joins, then for each $x\in L$, there is a largest $y\in L$ with $(x,y)\in E$. Therefore, let $E^\sharp:L\rightarrow L$ be the mapping defined by letting
$E^\sharp(x)$ denote the largest element with $(x,E^\sharp(x))\in E$.
Clearly, $x\leq E^\sharp(x)=E^\sharp(E^\sharp(x))$. Furthermore, if
$x\leq y$, then $y=x\vee y$, so since $(x,E^\sharp(x)),(y,E^\sharp(y))\in E$, we have $(E^\sharp(x)\vee E^\sharp(y),y)=(E^\sharp(x)\vee E^\sharp(y),x\vee y)\in E$. Therefore, $E^\sharp(x)\vee E^\sharp(y)\leq E^\sharp(y)$, hence $E^\sharp(x)\leq E^\sharp(y)$. Therefore, $E^\sharp$ is a closure operator.
If $L$ is a complete lattice and $E:L\rightarrow L$ is a closure operator, then $\ker(E)$ is a congruence that preserves arbitrary joins. If $C:L\rightarrow L$ is a closure operator, then $C=\ker(C)^\sharp$. If $E$ is a congruence that preserves arbitrary joins, then $E=\ker(E^\sharp)$. Therefore, the closure operators, closure systems, and the arbitrary join-preserving equivalence relations are all in a one-to-one correspondence with each other.
Adjoints
If $X,Y$ are posets, $f:X\rightarrow Y,g:Y\rightarrow X$ are mappings, and $f(x)\leq y$ iff $x\leq g(y)$, then we say that $f$ is a lower adjoint to $g$ and $g$ is an upper adjoint to $f$. Every function has at most one upper adjoint, and every function has at most one lower-adjoint. Every lower adjoint preserves all existing joins, and every upper adjoint preserves all existing meets. Furthermore, if $X,Y$ are complete lattices, then $f$ is a lower adjoint to some $g$ if and only if $f$ preserves arbitrary joins, and $g$ is an upper adjoint to some $f$ if and only if $g$ preserves arbitrary meets.
We say that a subset $D$ of a poset $X$ is an interior system if for each $x\in X$ there is a greatest $y\in D$ with $x\geq y$. We say that a function $D:X\rightarrow X$ is an interior operator if $x=D(x)=D(D(x))\geq D(y)$ whenever $x\geq y$. If $D:X\rightarrow X$ is an interior operator, then let $D^\circ=D[X]$ denote the dual interior system, and if $D\subseteq X$ is an interior operator, then let $D^\circ$ denote the interior system with $D^{\circ\circ}=D$.
If $f:X\rightarrow Y$ is left-adjoint to $g:Y\rightarrow X$, then $g\circ f$ is a closure operator, and $f\circ g$ is an interior operator. The mappings $f,g$ restrict to inverse order preserving bijections between the closure system $(g\circ f)^*$ and the interior system $(f\circ g)^\circ$. Furthermore, $f[X]=(f\circ g)^\circ$ and $g[Y]=(g\circ f)^*$, and $g\circ f=\ker(f)^\sharp$.
If $R\subseteq X\times Y,f=R^*$, and $g$ is the upper-adjoint of $f:P(X)\rightarrow P(Y)$, then $g(V)=R^{-1}[V^c]^c$ for each $V\subseteq Y$.
Here, the lattice $(f\circ g)^\circ=\{R[U]\mid U\subseteq X\}$ is analogous to the rank of the relation $R$, so define $\text{Rank}(R)=(f\circ g)^\circ$. We observe that the lattice $\text{Rank}(R)$ is isomorphic to $\{g(V)\mid V\subseteq Y\}=\{R^{-1}[V^c]^c\mid V\subseteq Y\}$ which is anti-isomorphic to $\text{Rank}(R)$ .
Quotient of matrix algebras
Let $\text{Rel}_n$ denote the set of all binary relations on the set $\{1,\dots,n\}$. Then define a mapping $\Phi:M_n([0,\infty))\rightarrow\text{Rel}_n$ by letting $\Phi((a_{i,j})_{i,j})=\{(j,i)\mid a_{i,j}>0\}$. Then $\phi$ is a semiring homomorphism in the sense that $\Phi(AB)=\Phi(B)\circ \Phi(A)$ for $A,B\in M_n([0,\infty))$. $M_n([0,\infty))$ contains all stochastic matrices, and I have used the antihomomorphism $\Phi$ in this answer to give examples of doubly stochastic matrices that cannot be factored into matrices of the form $(1-\lambda)P+\lambda Q$ for some permutation matrices $P,Q$.
