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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 peharps 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 $\mathbb{F}_2$.

Thanks for any help.

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 $\mathbb{F}_2$.

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 $$(AB)_{ik} = \bigvee_{j=1}^k (A_{ij} \land B_{jk}) ,$$ where $\lor$ and $\land$ are the logical "or" and "and" on $\mathbb{B}_2$. (and peharps the operations of matrix-or and matrix-and defined pointwise.)

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 $\mathbb{F}_2$.

Thanks for any help.

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 peharps 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 $\mathbb{F}_2$.

Thanks for any help.

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 operations of matrix multiplication $$(AB)_{ik} = \bigvee_{j=1}^k (A_{ij} \land B_{jk}) ,$$ where $\lor$ and $\land$ are the logical "or" and "and" on $\mathbb{B}_2$. (and peharps the operations of matrix-or and matrix-and defined pointwise.)

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 $\mathbb{F}_2$.

Thanks for any help.

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 $$(AB)_{ik} = \bigvee_{j=1}^k (A_{ij} \land B_{jk}) ,$$ where $\lor$ and $\land$ are the logical "or" and "and" on $\mathbb{B}_2$. (and peharps the operations of matrix-or and matrix-and defined pointwise.)

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 $\mathbb{F}_2$.

Thanks for any help.