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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeAug 2nd 2010
    • (edited Aug 2nd 2010)

    On variety of algebras appears the sentence “(This paragraph may be original research. Probably the concept does appear in the literature but under a different name.)”. The paragraph in question is about typed varieties of algebras. Looking at the history, this sentence (and indeed, the whole page!) appears to be due to Toby (Bartels).

    I’m curious as to what part that sentence refers to, in particular due to my interest in what I call graded varieties of algebras (nomenclature coming from algebraic topology and graded cohomology theories), which I thought was just an example of a heterogeneous variety of algebras, a term that I’ve come across in the literature. Certainly the concepts feel closely related, and it took a fair amount of paper chasing to find the term “heterogeneous” (though “many-sorted” theories seemed a bit more of a common term), but despite my interest, I’m no expert and am sure I’m missing something. Problem is: I don’t know what and I don’t know how to properly formulate my question!

    (Added in edit): Actually, I see that the term “multisorted” is in use on Lawvere theory.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeAug 2nd 2010

    That sentence is not a coy reference to something with a name that I know but which you are expected to guess. It just seemed like an obvious idea, but I had never read about it, and did not trust that the literature would follow my terminology.

    But now that you mention it, ‘many-sorted’ sounds like a good term to use.

    Maybe your question is whether what you’ve read about heterogeneous and many-sorted varieties of algebras matches what I wrote at variety of algebras? In that case, maybe you should say what the definitions are that you read, and then together we can see if they line up.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeAug 2nd 2010

    As is common in mathematics, the exact definition tends to be spread out and in confusing language, so here’s what I’ve subsumed from my reading. Any mistakes are my own!

    A many-sorted (or multi-sorted or heterogeneous or graded) algebra is one in which the underlying “thing” is not a set, but a graded set. In the hunting of the Hopf ring, we defined a graded set to be a functor from a certain fixed set (viewed as a discrete small category) into Set. Then operations and identities in this algebra, when interpreted as morphisms in the appropriate setting, have their inputs and outputs labelled by elements of the grading set. Other than that, everything is pretty much the same as in the single-sorted case. By my reading, this accords with what you wrote.

    In the hunting of the Hopf ring, this was part of section 2 on the background from general algebra. I’ve dug back in our references and remembered that Algebras with actions and automata, by Kuhnel, Pfender, Meseguer, and Sols, was one of our main sources for the ideas (though not the notation!). That’s: MR666494 in MathSciNet. I think it’s available online (if not, ask me).

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeAug 2nd 2010
    • (edited Aug 2nd 2010)

    By my reading, this accords with what you wrote.

    Mine too. Your graded set is simply the disjoint union of (the images of) my sorts.

    I think it’s available online (if not, ask me).

    It is (pdf).

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