Re: universal algebra and partial algebra
Ronnie Brown asks -
Is it naive to ask that universal algebra theory should also include partial algebras, and in particular groupoids, categories, multiple groupoids, multiple categories (of various kinds)?
It seems that where the methods of universal algebra generalize well to cover partial algebras is in _essentially_ algebraic theories, i.e. those in which there is a hierarchy of operators such that the domain of definition of each is defined by a conjunction of equations involving more primitive operators. This certainly covers categories and groupoids - anyone who knows Phillip Higgins' book can see how smoothly universal algebra works for them. As for multiple groupoids and multiple categories, I can't say because I'm not familiar with them. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Is it naive to ask that universal algebra theory should also include partial algebras, and in particular groupoids, categories, multiple groupoids, multiple categories (of various kinds)?
It seems that where the methods of universal algebra generalize well to cover partial algebras is in _essentially_ algebraic theories, i.e. those in which there is a hierarchy of operators such that the domain of definition of each is defined by a conjunction of equations involving more primitive operators. This certainly covers categories and groupoids - anyone who knows Phillip Higgins' book can see how smoothly universal algebra works for them. As for multiple groupoids and multiple categories, I can't say because I'm not familiar with them. As far as the partial case is concerned: why not generalized algebraic theories ala Cartmell? They seem to have essentially the same basic semantic interpretation (categories with finite limits) and are applicable to similar examples. I prefer generalized algebraic theories because they are essentially a first order subset of Type Theory (ala Martin L"of) which is used by many people in Computer Science to formalize mathematics and to verify algorithms. I personally also find generalized algebra more natural, e.g. when formalizing catgeories (why not call them monoidoids ?) one introduces homsets and composition as indexed by the objects instead of having cod and dom and composition as a partial operation. Thorsten Altenkirch Host: muppet77.cs.chalmers.se Office: 2438 snail: Dep. of Computer Sciences; Chalmers U. of Technology; 412 96 Gothenburg phone: (+ 46) 31 772 1079 [office], (+ 46) 31 55 54 40 [home] Sweden ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I just wanted to add that it is unneccessary, in defining essentially algebraic theories to worry about a hierarchy of types in which the domain of each is defined equationally in terms of simpler types. I realized this because I once tried to show that the theories classified by essentially algebraic theories are essentially weaker than those classified by finite limit sketches. They are not because if you had two types, the domain of each of which was described equationally in terms of the other (or a type whose domain was described equationally in terms of itself), you could add new dummy types at the lowest level of the hierarchy, use them to define the domains of the type(s) in question and then add new unary operations of each of the dmmy types taking values as the other and finally equations that forced those operations to be isomorphisms. Nonetheless, I think that FL sketches are the way to go and the definition of essentially algebraic a kludge. Michael ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (3)
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barr＠triples.Math.McGill.CA -
sjv＠doc.ic.ac.uk -
Thorsten Altenkirch