These are all well-known things in simplicial geometry [1] which is used everywhere in computational topology and discrete differential geometry. It turns out all of these operations can be done efficiently by considering a matrix over F_2 and considering its several forms. These manipulations lead to a natural (computable!) discretization of differential/topological methods because they're fully constructive and have beautiful theorems. I'd highly recommend looking into it if you're interested.
There may even be a nice way of taking elements from the algebra and extending these definitions to fit this idea of '∂' or closure or interior, etc; but I have little intuition for what the algebra might look like and whether it would admit nice expressions of this (personally, I was more motivated by the possible connection to computability theory and languages). Once I have some time, I may sit down and toy around with the idea.
There may even be a nice way of taking elements from the algebra and extending these definitions to fit this idea of '∂' or closure or interior, etc; but I have little intuition for what the algebra might look like and whether it would admit nice expressions of this (personally, I was more motivated by the possible connection to computability theory and languages). Once I have some time, I may sit down and toy around with the idea.
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[1] https://en.wikipedia.org/wiki/Simplicial_complex