I have an algebraic structure that I'll define with the ordered triple $(\{0, 1 \}, \oplus, \times)$ such that the following properties hold:
$$ 0 \oplus 0 = 0$$ $$ 1 \oplus 1 = 1 $$ $$0 \oplus 1 = 1 \oplus 0 = 1$$
Multiplication given by $\times$ holds as it would normally for the integers $0$ and $1$, and thus needs no distinction.
I already know that $\oplus$ is associative (as can be verified by a few simple computations).
The trick comes from using $\oplus$ in substitution of the usual operation of addition when multiplying vectors by matrices and matrices by other matrices. Both vectors and matrices will only have elements of $\{ 0, 1 \}$ in them.
To wit: given a vector $v \in \{ 0, 1\}^n$ and some square matrix $M \in \{ 0, 1\}^{n \times n}$, I want to define the following as such:
$$ (M v)_{i} = \bigoplus_j M_{ij} \times v_j$$ and
$$ (M_1 M_2)_{ij} = \bigoplus_k M_{ik} \times M_{kj}$$
Normally we use a base field $\mathbb{F}$ with the usual operation of addition, but for this case I only need two properties to hold for this algebraic structure to serve its purpose:
$$ M_1 (M_2 v) = (M_1 M_2)v$$ and $$ M_1(M_2 M_3) = (M_1 M_2) M_3$$
That is, I need to know that I have the usual properties of associativity when multiplying (I have no need for group properties like existence and uniqueness of inverse, though).
Instead of going into the gory details, are there some clean or economical tools out there that quickly and concisely guarantee I have associativity?
Thank you in advance for the help. For those curious: the square matrices that I am using this custom operation in are generally not invertible even when using a field $\mathbb{F}$. Additionally, this question is related to my research, and is not some odd exercise dug out of a textbook.