Abstract
In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.
This research was supported by the Science Fund of the Republic of Serbia, Grant No. 7750027: Set-theoretic, model-theoretic and Ramsey-theoretic phenomena in mathematical structures: similarity and diversity – SMART
Communicated by Miroslav Ploščica
References
[1] Adámek, J.—Herrlich, H.—Strecker, G. E.: Abstract and Concrete Categories: The Joy of Cats, Dover Books on Mathematics, Dover Publications, 2009.Search in Google Scholar
[2] Dasilva Barbosa, K.: A categorical notion of precompact expansions, preprint (arXiv:2002.11751) 2020.Search in Google Scholar
[3] Deuber, W.: A generalization of Ramsey’s theorem for regular trees, J. Combin. Theory, Ser. B 18 (1975), 18–23.10.1016/0095-8956(75)90057-XSearch in Google Scholar
[4] Deuber, W.—Rothschild, B. L.: Categories without the Ramsey property. In: Colloq. Math. Soc. János Bolyai, 18, Vol. I, North-Holland, Amsterdam-New York, 1978, pp. 225–249.Search in Google Scholar
[5] Graham, R. L.—Rothschild, B. L.: Ramsey’s theorem for n-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292.10.1090/S0002-9947-1971-0284352-8Search in Google Scholar
[6] Graham, R. L.—Leeb, K.—Rothschild, B. L.: Ramsey’s theorem for a class of categories, Adv. Math. 8 (1972), 417–433.10.1016/0001-8708(72)90005-9Search in Google Scholar
[7] Kechris, A. S.—Pestov, V. G.—Todorčević, S.: Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups, GAFA Geom. Funct. Anal. 15 (2005), 106–189.10.1007/s00039-005-0503-1Search in Google Scholar
[8] Leeb, K.: The Categories of Combinatorics. Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970.Search in Google Scholar
[9] Leeb, K.: Vorlesungen über Pascaltheorie, Arbeitsberichte des Instituts für mathematische Maschinen und Datenverarbeitung, Band 6, Nummer 7, Firedrich Alexander Universität, Erlangen, 1973.Search in Google Scholar
[10] Mašulović, D.: Pre-adjunctions and the Ramsey property, European J. Combin. 70 (2018), 268–283.10.1016/j.ejc.2018.01.006Search in Google Scholar
[11] Mašulović, D.: The Kechris-Pestov-Todorčević correspondence from the point of view of category theory, Appl. Categ. Structures 29 (2021), 141–16910.1007/s10485-020-09611-zSearch in Google Scholar
[12] Mašulović, D.: Ramsey degrees: big v. small, European J. Combin. 95 (2021), Art. 10332310.1016/j.ejc.2021.103323Search in Google Scholar
[13] Mašulović, D.: Dual Ramsey properties for classes of algebras, European J. Combin. 112 (2023), Art. 103716.10.1016/j.ejc.2023.103716Search in Google Scholar
[14] Mašulović, D.—Scow, L.: Categorical equivalence and the Ramsey property for finite powers of a primal algebra, Algebra Universalis 78 (2017), 159–179.10.1007/s00012-017-0453-0Search in Google Scholar
[15] Müller, M.—Pongrácz, A.: Topological dynamics of unordered Ramsey structures, Fund. Math. 230 (2015), 77–98.10.4064/fm230-1-3Search in Google Scholar
[16] Nguyen van Thé, L: More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fund. Math. 222 (2013), 19–47.10.4064/fm222-1-2Search in Google Scholar
[17] Oates-Williams, S.: Ramsey varieties of finite groups, European J. Combin. 9 (1988), 369–373.10.1016/S0195-6698(88)80067-2Search in Google Scholar
[18] Prömel, H. J.: Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177–208.10.1016/0097-3165(85)90036-6Search in Google Scholar
[19] Ramsey, F. P.: On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.10.1112/plms/s2-30.1.264Search in Google Scholar
[20] Sokić, M.: Semilattices and the Ramsey property, J. Symb. Log. 80 (2015), 1236–1259.10.1017/jsl.2014.40Search in Google Scholar
[21] Sokić, M.: Unary functions, European J. Combin. 52 (2016), 79–94.10.1016/j.ejc.2015.09.003Search in Google Scholar
[22] Voigt, B.: The partition problem for finite abelian groups, J. Combin. Theory Ser. A 28 (1980), 257–271.10.1016/0097-3165(80)90069-2Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets
Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets