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I don't think it's a lattice per se, but Darcs had a notion of an "algebra of patches": http://darcs.net/Theory/GaneshPatchAlgebra. Pijul does something similar, claiming to have "a formally correct version of Darcs' theory of patches" https://pijul.org/manual/why_pijul.html

I've never used Pijul, but I miss Darcs almost every day when interacting with Git.

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Based on that description of Pijul, it does seem to be a semilattice: it has a merge operation that's associative and commutative, and while it doesn't explicitly state that it's idempotent it's hard to imagine merging two states and getting anything else back than the same state.

Not super familiar with Pijul though!

I am one of the authors of Pijul. It is indeed a semilattice, but not exactly "commutative": the only operations that commute are operations that could be done in parallel. For example, adding a file doesn't commute with editing it.

But unlike in Darcs, where conflicts are messy and not super well-defined in all cases (and not to mention, sometimes cause Darcs to run for a time exponential in the size of the repo's history), two conflicting edits always commute in Pijul.

The main points of Pijul were (1) to solve the soundness issues with conflicts and (2) to fix the exponential problems.

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