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I am still learning this stuff (and have only had undergrad abstract algebra), so thank you for the context and motivation.

As it happens, today I was thinking about semirings, and how to apply them to discrete event systems[1][2], and was fascinated to see that both algebraic geometers (Alexander Grothendieck) and computer scientists (Robin Milner) were doing[3] some of the same[4] sorts of tricks to transfer algebraic structures (like rings) over to settings lacking an inverse (like semirings)--rather than take the direct approach (as you mentioned in your lattice example):

There are many times in mathematics where some class of objects has a nice additive structure, except that one can’t always invert elements. One approach is to say, “oh well,” and develop the theory of monoids; another,which we’ll discuss today, adds in formal inverses, in effect pretending that they exist, and uses them to recover something useful.

[1] https://www.cl.cam.ac.uk/%7Esd601/papers/semirings.pdf

[2] http://r6.ca/blog/20110808T035622Z.html

[3] https://web.ma.utexas.edu/users/a.debray/lecture_notes/sophe...

[4] https://old.reddit.com/r/haskell/comments/2d8q24/additivemul...

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