Let $A_{n\times n}=(a_{ij}),B_{n\times n}=(b_{ij}) \in M_{n}(\mathbb{R})$, where $a_{ij},b_{ij} \in \lbrace 0,1\rbrace$. Boolean sum of $A,B$ denoted by $(A \oplus B)_{n\times n}=(a_{ij}\oplus b_{ij})$ is the matrix in $M_{n}(\mathbb{R})$ such that $0\oplus 0 = 0$, $0\oplus 1 = 1$, $1\oplus 0=1$ and $1\oplus 1 =1$. Is there any inequality or relation between eigenvalues of $A,B$ and $A\oplus B$? (specially, when $A,B$ are symmetric)

The operation is actually elementwise maximum, and the matrices are both nonnegative, so PerronFrobenius theorem together with the RayleighRitz characterization of maximum (perronfrobenius) eigenvalue seems to indicate that $\lambda_\max(A\oplus B) \geq \max \{\lambda_\max(A), \lambda_\max(B)\}.$ I don't know why you would use the $\oplus$ notation for this. 

