Finite implications and infinite models
In this chapter we discuss implications that hold for finite magmas but not infinite ones. The \(820\) such implications between laws of order up to \(4\) are listed at https://github.com/teorth/equational_theories/blob/main/data/Austin_implications.txt. The implications are proven using consequences of finiteness such as the equivalence of injectivity and surjectivity. The non-implications for infinite magmas are obtained by constructing infinite models, which are more challenging to construct—they can't be proved by directly giving an operation table and checking which laws it satisfies. Here we simply build a few of these infinite counterexamples, while in Infinite magma constructions we describe thoroughly the construction techniques developed throughout this project.
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