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In his dissertation on "Functorial semantics of algebraic theories", Lawvere says in his introduction that

"from the category (or more precisely from an underlying-set functor) we can recover, not only the identities which hold between given operations in a class of algebras, but also the operations themselves"

The construction is based on the "algebraic structure" functor, adjoint to the "semantics" functor, which assigns to each algebraic theory the category of algebras of which it is a theory.

Is it possible to use this machinery to find THE right presentation of, for example, the theory of groups or Boolean algebras, among the infinite possibilities of presentations?

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    $\begingroup$ Considering the concept that bears his name, you may be unsurprised to learn that the presentation so obtained is the Lawvere theory of the given category of algebras, which though completely canonical and natural, can hardly be said to be a presentation useful for calculations... $\endgroup$
    – Zhen Lin
    Commented Nov 8 at 15:31
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    $\begingroup$ How are you defining what it means for a presentation to be the 'right' one? $\endgroup$
    – James E Hanson
    Commented Nov 8 at 16:19
  • $\begingroup$ @zhen-lin : it's well known that there are different axiomatic systems for groups or boolean algebras (cup, cap, minus ; cup, minus, zero ; etc). I wondered if Lawvere-theoretical machinery allows to define one of these lists of primitive operations as the right one, the good definition of boolean algebras. $\endgroup$
    – Sylvain Cabanacq
    Commented Nov 9 at 9:47
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    $\begingroup$ Yes: there is a canonical one, namely the one that has all the operations. $\endgroup$
    – Zhen Lin
    Commented Nov 9 at 11:43
  • $\begingroup$ The title asks about presentations of groups, but the question seems to be about presentations of the theory of groups. These are very different things -- perhaps the title should be altered to something less misleading? $\endgroup$
    – HJRW
    Commented Nov 10 at 13:22

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