universal algebra aand partial algebra
Re the discussion on empty algebras: 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)? In algebraic work with these objects, the empty object is considered as a matter of course, since one wants the theory of groupoids, say, to be an extension in some sense of the theory of sets. I expect that such a notion of extension is standard in universal algebra. Readers will be interested in the following quotation from p.4 of Ch I of A Connes "Non commutative geometry" (an English translation of "G\'eometrie non commutative", with many additions, about three times as long, to be published by Academic Press). "It is fashionable among mathematicians to despise groupoids [not among readers of this bulletin!] and consider that only groups have an authentic mathematical status, probably because of the pejorative suffix oid. To remove this prjudice we start Chapter I by Heisenberg's discovery of quantum mechanics. The reader will hopefully realise there how the experimental results of spectroscopy forced Heisenberg to replace the classical frequency group of the system by the groupoid of quantum transitions. Imitating for this groupoid the construction of the group convolution algebra Heisenberg rediscovered matrix multiplication and invented Quantum mechanics." Ronnie Brown ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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MAS010@BANGOR.AC.UK