Here are some remarks in connection with the Universal Algebra versus Category Theory debate initiated by Saunders MacLane. (i) There is yet another way to describe a (multisorted) algebraic theory, alias clone, namely as a Cartesian multicategory, that is, a Gentzen style deductive system with appropriate equations between deductions. Deductions f:A_1...A_n ---> B should be viewed as operations, in the spirit of the formulas as types paradigm. (ii) Why anyone would deny that the subalgebras of an algebra form a complete lattice is beyond me. The subobject lattice and congruence lattice are both subsumed in the lattice of subcongruences, which I found myself in my paper "Goursat's theorem and the Zassenhaus lemma", Can. J. Math. 10(1957), 45-56. Subcongruences have recently been resurrected as partial equivalence relations; they can of course be defined in any category. (iii) If one studies algebras on graphs instead of sets, as pioneered by Burroni, one may view many structured categories, even toposes, as algebras. (iv) To avoid problems with empty types, e.g. lack of transitivity, one should declare variables both in equations and deductions. Phil Scott and I did so in our book of 1986, by placing a subscript X={x_1,...x_n} on the equality and entailment symbols. Unfortunately, some other books did not follow this practice. Jim Lambek ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Some people may be interested in reading what Jose Meseguer and I did with variables in equations and deduction, starting around 1981. The final 1985 version discusses some connections with categorical approaches; for example, we point out a flaw in Benabou's work. All this was motivated by the need to reason about abstract data types in Computer Science. @article(complas, title = "Completeness of Many-sorted Equational Logic", author = "Joseph Goguen and Jos\'e Meseguer", year = 1985, journal = "Houston Journal of Mathematics", volume = 11, number = 3, pages = "307--334", note = "Preliminary versions have appeared in: {\it SIGPLAN Notices}, July 1981, Volume 16, Number 7, pages 24--37; SRI Computer Science Lab, Report CSL-135, May 1982; and Report CSLI-84-15, Center for the Study of Language and Information, Stanford University, September 1984") &&&&&&&&&&&&&&&&&&&&&&&&&&&&& Signature File &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Joseph A. Goguen, Professor of Computing Science, Programming Research Group, Oxford University Computing Lab, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. email: Joseph.Goguen@prg.ox.ac.uk [internet] -- usually also works in the UK, but if not, try Joseph.Goguen@uk.ac.ox.prg phone: 283504 [my office]; 283505 [secy]; 273838 [PRG office]; 273839 [FAX].
From USA, dial 011-44-865-...; from UK, dial (0865)-... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (2)
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es@math.mcgill.ca -
Joseph.Goguen@prg.oxford.ac.uk