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The independence of regular identities in groupoids. (English) Zbl 0699.20044

We say that the conditions of a groupoid identity \(f=g\) are independent over a set A if for each \(<a_ 1,...,a_ n>\in A^ n\) there is a binary operation \(\circ\) on A such that in the groupoid \({\mathfrak A}=<A,\circ >\) we have \(f_{{\mathfrak A}}(a_{i_ 1},...,a_{i_ k})\neq g_{{\mathfrak A}}(a_{j_ 1},...,a_{j_ l})\) but \(f_{{\mathfrak A}}(b_{i_ 1},...,b_{i_ k})=g_{{\mathfrak A}}(b_{j_ 1},...,b_{j_ l})\) for any \(<b_ 1,...,b_ n>\in A^ n\setminus \{<a_ 1,...,a_ n>\}\). The author generalizes the independence of identities to weak independence. Then regular groupoid identities (i.e. identities having the same set of variables on both sides) are investigated from the weak independence point of view.
Reviewer: J.Duda

MSC:

20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
08A40 Operations and polynomials in algebraic structures, primal algebras
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