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I am embarrassed not to have recognized this sooner, but a 'canonical' structure that models Infinite Craft is the Commutative Magma — definitionally, a set $S$ with a binary operation $\oplus$ that is closed ($s\oplus t$ is defined and is $\in S$ for all $s,t\in S$) and commutative. Unfortunately, the level of generality at play here means that this is not necessarily helpful for practical considerations, but those are the magic words I'd start with to try and find more information if it's out there.