Abstract
This article describes well quasi orders as a category, focusing on limits and colimits. In particular, while quasi orders with monotone maps form a category which is finitely complete, finitely cocomplete, and with exponentiation, the full subcategory of well quasi orders is finitely complete and cocomplete, but with no exponentiation. It is interesting to notice how finite antichains and finite proper descending chains interact to induce this structure in the category: in fact, the full subcategory of quasi orders with finite antichains has finite colimits but no products, while the full subcategory of well founded quasi orders has finite limits but no coequalisers. Moreover, the article characterises when exponential objects exist in the category of well quasi orders and well founded quasi orders. This completes the systematic description of the fundamental constructions in the categories of quasi orders, well founded quasi orders, quasi orders with finite antichains, and well quasi orders.
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References
Borceux, F.: Handbook of Categorical Algebra 1: Basic Category Theory. Encyclopedia of Mathematics and Its Applications, vol. 50. Cambridge University Press, Cambridge (1994)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors. Am. J. Math. 35(4), 413–422 (1913)
Gallier, J.H.: What’s so special about Kruskal’s theorem and the ordinal \(\Gamma _{o}\)? A survey of some results in proof theory. Ann. Pure Appl. Log. 53(3), 199–260 (1991)
Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley, New York (1990)
Kruskal, J.B.: The theory of well-quasi-ordering: a frequently discovered concept. J. Comb. Theory (A) 13, 297–305 (1972)
Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, Berlin (1997)
Milner, E.C.: Basic WQO- and BQO-theory. In: Rival, I. (ed.) Graphs and Order, NATO ASI Series, vol. 147, pp. 487–502. Springer, Berlin (1985)
Seisenberger, M.: An inductive version of Nash–Williams’ minimal-bad-sequence argument for Higman’s lemma. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) Types for Proofs and Programs (TYPES’00). Lecture Notes in Computer Science, vol. 2277, pp. 233–242. Springer, Berlin (2001)
Acknowledgements
Thiswork has been supported by the project Correctness byConstruction (CORCON), EU 7 th framework programme, Grant No. PIRSES-GA-2013-612638, European Union Research Agency, and by the project Abstract Mathematics for Actual Computation: Hilbert’s Program in the 21st Century, John Templeton Foundation.
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Benini, M., Bonacina, R. Well quasi orders in a categorical setting. Arch. Math. Logic 58, 501–526 (2019). https://doi.org/10.1007/s00153-018-0649-9
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DOI: https://doi.org/10.1007/s00153-018-0649-9