In case anyone missed it, Peter Smith has written an excellent little tutorial on Galois connections. It starts with the needed order theory, then moves on to the specifics of Galois connections with an a nice application to explaining the link between theories and their models. Reading it has given me a little bit of motivation for writing another post on gaggles. Reading Peter’s tutorial helped me see how gaggles work. This probably should’ve been obvious: they are built on Galois connections. Each gaggle has an n+1-ary relation that provides the backbone of the semantics for its family of n-ary operations. Dunn’s proofs are for gaggles whose operations are of arbitrary arities. This raises a question of what the corresponding Galois connections are like. This seems like a question because the Galois connections involve two functions, which is natural enough for families of binary operations. For, say, an 8-ary operation, do we need a daisy chain of 8 functions to set up a general Galois connection? Dunn should have something to say about this.

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