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  1. defining arity of a function


    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 6th 2021

    Given a cardinal number nn, an n-ary operation on a set SS is a function i:[n]() i:S nS\prod_{i:[n]} (-)_i \colon S^n \to S from the cartesian power S nS^n to SS, where [n] is a set with nn elements. The arity of the operation is nn.

    How are we to read i:[n]() i:S nS\prod_{i:[n]} (-)_i \colon S^n \to S? Am I being slow?

    Also, we need ’arity’ for relations. E.g., at signature (in logic) we have

    A set Rel(Σ)Rel(\Sigma) whose elements are called relation symbols, equipped with a function ar:Rel(Σ)S *ar: Rel(\Sigma) \to S^* to the free monoid on SS which prescribes an arity for each relation symbol,

    • CommentRowNumber3.
    • CommentAuthorvarkor
    • CommentTimeMay 6th 2021

    How are we to read i:[n]() i:S nS\prod_{i:[n]} (-)_i \colon S^n \to S? Am I being slow?

    I am also confused. An n-ary operation on a set SS is just a function S nSS^n \to S.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2021

    I guess it’s just a typo coming from a change of mind between writing S nS^n or i[n]S\underset{i \in [n]}{\prod} S.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    I have fixed it. Also in the Properties-section further below.

    diff, v2, current

  2. linking to operad


    diff, v5, current

  3. added section on finitary operations


    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2024

    have slightly expanded at the point where the term “finitary groupoid” appears, for clarification.

    diff, v7, current