Comments on: Arguesian Lattices https://www.arsmathematica.net/2014/09/23/arguesian-lattices/ Dedicated to the mathematical arts. Fri, 29 May 2015 09:17:44 +0000 hourly 1 https://wordpress.org/?v=4.1.41 By: Walt https://www.arsmathematica.net/2014/09/23/arguesian-lattices/#comment-85096 Wed, 01 Oct 2014 21:32:41 +0000 http://www.arsmathematica.net/?p=1732#comment-85096 Thanks for the references. That’s surprising, because the theory of modular lattices is undecidable. I was wondering if the theory of sublattices of modules (or in general the theory of congruence lattices of congruence modular varieties that you mention) is decidable, and I was thinking about asking on Math Overflow. I should check your references first.

I have the impression that the complete list of the identities satisfied in these cases is not yet known.

]]> By: Todd Trimble https://www.arsmathematica.net/2014/09/23/arguesian-lattices/#comment-85091 Wed, 24 Sep 2014 11:54:35 +0000 http://www.arsmathematica.net/?p=1732#comment-85091 Whenever you have a lattice of commuting equivalence relations, as is the case for lattices of submodules, or more generally lattices of submodels of a model of a Mal’cev algebraic theory, you have an infinite list of modular-law type identities of which the arguesian law is but one. This isn’t an area of expertise of mine, but you can learn more from this article courtesy of Google books, and also in Freyd and Scedrov’s Categories, Allegories, section 2.158. Freyd and Scedrov indicate that the theory of lattices satisfying all of these modular-type identities is decidable, and to me it smells very similar to the calculus developed by Rota and his collaborators, but that there are close connections between these works should be treated for now as a hunch.

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