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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 13th 2015

added references to essentially algebraic theory. Also equipped the text with a few more hyperlinks.

• CommentRowNumber2.
• CommentAuthorspitters
• CommentTimeOct 12th 2018

• redirected “Cartesian theory” here
• added brief definition of cartesian theory.

Steve Vickers

1. Added “cartesian logic” as related topic.

“Cartesian theory” really ought to be redirected here, as the notions are equivalent. I am attempting to do that.

Steve Vickers

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 18th 2019

• CommentRowNumber6.
• CommentAuthoratmacen
• CommentTimeSep 28th 2019

In particular, one selling point of intrinsic syntax seems to be its essentially-algebraic character

I don’t understand the details of the connection between essentially algebraic theories and type systems. Which type systems are essentially algebraic? Is it exactly those with intrinsic syntax? How do I rigorously tell when syntax is “intrinsic”, in an arbitrary metalanguage?

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 19th 2020

Transferred material from cartesian theory > history.

• CommentRowNumber8.
• CommentAuthoranuyts
• CommentTimeNov 23rd 2021
• (edited Nov 23rd 2021)
In the traditional syntactic definition, to me it is not immediately obvious that it is always sufficient to have only equations using total operations, and no clear reference is included. However, I think I have come up with a transformation from EATs that use non-total operations in their equations to systems that do not.

Assume we have an operator
σ : {x : ∏_i s_i | P(x)}.
Then we can add a sort P* and a total operation outP* : P* -> ∏_i s_i and include P(outP*(p)) in the equational theory of the system. Then σ can be rewritten as
σ : {(x, p) : (∏_i s_i) x P* | outP*(p) = x}.
I guess we also want to express in the equational theory that if p, q : P*, then p = q. We probably also want to make sure that if P(x) then x = outP*(p) for some p, but I do not see how to do that, since P mentions non-total operations.

If this is indeed the idea, then I suggest to include this in the article.
• CommentRowNumber9.
• CommentAuthoranuyts
• CommentTimeNov 23rd 2021
To answer atmacen 1.5 years post datum: Cartmell shows that EATs and GATs are equivalent. See https://ncatlab.org/nlab/show/generalized+algebraic+theory
If we include simultaneous substitutions as a syntactic sort, then type theories can be formulated as GATs (or perhaps even multisorted ATs in the case of simple type theories).
• CommentRowNumber10.
• CommentAuthoranuyts
• CommentTimeFeb 14th 2023

Cite keml-diagrams.