I was studying this basis defined for the digital plane topology and started wondering if there is a way to connect/explain the basis in terms of an algebraic structure?
The collection $\mathcal{B}_{p} = \left \{B(m,n) \mid \ m,n \in \mathbb{Z} \ X \ \mathbb{Z}\right\}$ is a basis for the digital plane topology.
- $\left\{(m,n)\right\} \ m,n \ odd$
- $\{(m+a,n) \mid a=-1,0,1\}$ $\ m$ is even, $n$ is odd
- $\{(m,n+b) \mid b=-1,0,1\}$ $\ m$ is odd, $n$ is even
- $\{(m+a,n+b) \mid a,b=-1,0,1\} \ m,n$ are both even
It's just the product topology of the product of the standard digital line $L$ with itself, where $L$ is just the set $\mathbb{Z}$ with the base $\mathcal{B}_L$ consisting of all sets $\{n\}$ for $n$ odd and $\{n-1, n, n+1\}$ for $n$ even. The standard product base w.r.t. $\mathcal{B}_L$ is the base you described.
$L$ cannot be made into a topological group because it's not homogenous: there are no self-homeomorphisms of the space mapping even to odd numbers e.g. This probably means that also weaker continuous operations will be ruled out (semigroup?), both on $L$ and the digital plane $L \times L$. I certainly don't know about any algebraic connection.