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I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^m$ to $\mathbb{B}^n$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

EDIT: Here is Löwenheim's theorem I mention above: In a Boolean algebra $(\mathbb{B},\cup,.,',0,1)$ (I guess that $.$ is the intersection and $'$ the negation), to each $(b_1,\dots,b_m)\in \mathbb{B}^m$, we can associate the system of equations $$\bigcup_{j=1}^na_{ij}x_j=b_i \ \ \ (i=1,\dots,m).$$ Löwenheim proved in a 1919 paper that the system is consistent (I guess that that means that there is a solution) for a given $(b_1,\dots,b_m)\in \mathbb{B}^m$ if and only if $$b_i\leq \bigcup_{j=1}^n a_{ij} \prod_{h=1,h\neq i}^m (a'_{hj}\cup b_h)\ \ \ (i=1,\dots,m).$$ I guess that $\leq$ means the inclusion (it is not explained in the paper). In the same paper, the author calls $x+y=xy'\cup x'y$ the ring sum which is "xor".

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^n$ to $\mathbb{B}^m$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

EDIT: Here is Löwenheim's theorem I mention above: In a Boolean algebra $(\mathbb{B},\cup,.,',0,1)$ (I guess that $.$ is the intersection and $'$ the negation), to each $(b_1,\dots,b_m)\in \mathbb{B}^m$, we can associate the system of equations $$\bigcup_{j=1}^na_{ij}x_j=b_i \ \ \ (i=1,\dots,m).$$ Löwenheim proved in a 1919 paper that the system is consistent (I guess that that means that there is a solution) for a given $(b_1,\dots,b_m)\in \mathbb{B}^m$ if and only if $$b_i\leq \bigcup_{j=1}^n a_{ij} \prod_{h=1,h\neq i}^m (a'_{hj}\cup b_h)\ \ \ (i=1,\dots,m).$$ I guess that $\leq$ means the inclusion (it is not explained in the paper). In the same paper, the author calls $x+y=xy'\cup x'y$ the ring sum which is "xor".

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^m$ to $\mathbb{B}^n$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

EDIT: Here is Löwenheim's theorem I mention above: In a Boolean algebra $(\mathbb{B},\cup,.,',0,1)$ (I guess that $.$ is the intersection and $'$ the negation), to each $(b_1,\dots,b_m)\in \mathbb{B}^m$, we can associate the system of equations $$\bigcup_{j=1}^na_{ij}x_j=b_i \ \ \ (i=1,\dots,m).$$ Löwenheim proved in a 1919 paper that the system is consistent (I guess that that means that there is a solution) for a given $(b_1,\dots,b_m)\in \mathbb{B}^m$ if and only if $$b_i\leq \bigcup_{j=1}^n a_{ij} \prod_{h=1,h\neq i}^m (a'_{hj}\cup b_h)\ \ \ (i=1,\dots,m).$$ I guess that $\leq$ means the inclusion (it is not explained in the paper). In the same paper, the author calls $x+y=xy'\cup x'y$ the ring sum which is not "xor" either.

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^m$ to $\mathbb{B}^n$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

EDIT: Here is Löwenheim's theorem I mention above: In a Boolean algebra $(\mathbb{B},\cup,.,',0,1)$ (I guess that $.$ is the intersection and $'$ the negation), to each $(b_1,\dots,b_m)\in \mathbb{B}^m$, we can associate the system of equations $$\bigcup_{j=1}^na_{ij}x_j=b_i \ \ \ (i=1,\dots,m).$$ Löwenheim proved in a 1919 paper that the system is consistent (I guess that that means that there is a solution) for a given $(b_1,\dots,b_m)\in \mathbb{B}^m$ if and only if $$b_i\leq \bigcup_{j=1}^n a_{ij} \prod_{h=1,h\neq i}^m (a'_{hj}\cup b_h)\ \ \ (i=1,\dots,m).$$ I guess that $\leq$ means the inclusion (it is not explained in the paper). In the same paper, the author calls $x+y=xy'\cup x'y$ the ring sum which is "xor".

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^m$ to $\mathbb{B}^n$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in the case $m=n$ in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) and also in the case $m \neq n$ ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by $\mathbb{B}$. What I need to know is a necessary and sufficient condition for an application from $\mathbb{B}^m$ to $\mathbb{B}^n$ to be one-to-one. I read in the paper "Linear Boolean Equations and Generalized Minterms" by S. Rudeanu (Discrete Math 43 (1983) 241-248) that Löwenheim proved some theorems in a paper written in 1919. Hence my question :

Is there any more recent reference about this subject (systems of equations in Boolean algebra) ? And where can I find a proof of Löwenheim's theorem (that could help to understand) ?

All references I can found in some papers I can find (with difficulty) on the Web are unavailable online, and unavailable in my library.

EDIT: Here is Löwenheim's theorem I mention above: In a Boolean algebra $(\mathbb{B},\cup,.,',0,1)$ (I guess that $.$ is the intersection and $'$ the negation), to each $(b_1,\dots,b_m)\in \mathbb{B}^m$, we can associate the system of equations $$\bigcup_{j=1}^na_{ij}x_j=b_i \ \ \ (i=1,\dots,m).$$ Löwenheim proved in a 1919 paper that the system is consistent (I guess that that means that there is a solution) for a given $(b_1,\dots,b_m)\in \mathbb{B}^m$ if and only if $$b_i\leq \bigcup_{j=1}^n a_{ij} \prod_{h=1,h\neq i}^m (a'_{hj}\cup b_h)\ \ \ (i=1,\dots,m).$$ I guess that $\leq$ means the inclusion (it is not explained in the paper). In the same paper, the author calls $x+y=xy'\cup x'y$ the ring sum which is not "xor" either.