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Minimality of the system of seven equations for the category of finite sets. (English) Zbl 0903.18001

Summary: A. Burroni [Theor. Comput. Sci. 115, No. 1, 43-62 (1993; Zbl 0791.08004)], and later Y. Lafont [Lond. Math. Soc. Lect. Note Ser. 177, 191-201 (1992; Zbl 0789.18004)], proposed a presentation of the monoidal category of finite sets with three generators and seven equations. We prove that none of these equations is superfluous by considering interpretations into monoidal categories.

MSC:

18B05 Categories of sets, characterizations
68Q42 Grammars and rewriting systems
68Q70 Algebraic theory of languages and automata
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18C10 Theories (e.g., algebraic theories), structure, and semantics
18B20 Categories of machines, automata
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References:

[1] Burroni, A., Higher dimensional word problem, Theoret. Comput. Sci., 115, 43-62 (1993) · Zbl 0791.08004
[2] Lafont, Y., Penrose diagrams and 2-dimensional rewriting, (Fourman, M. P.; Johnstone, P. T.; Pitts, A. M., Applications of Categories in Computer Science. Applications of Categories in Computer Science, LMSLNS, Vol. 177 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 191-201 · Zbl 0789.18004
[3] Lafont, Y., Equational reasoning with 2-dimensional diagrams, (Lecture Notes in Computer Science, Vol. 909 (1995), Springer: Springer Berlin), 170-195
[4] Mac Lane, S., Categories for the Working Mathematician, (GTM, 5 (1971), Springer: Springer Berlin) · Zbl 0906.18001
[5] Penrose, R.; Rindler, W., Spinors and Space-time, Vol. 1: Two-spinor Calculus and Relativistic Fields (1986), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0602.53001
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