Structures with Two Partial Orders and the Amalgamation Property

Paolo Lipparini

doi:10.18778/0138-0680.2026.17

Authors

Paolo Lipparini Università di Roma “Tor Vergata”, Dipartimento di Amalgamatematica, Rome, Italy Author https://orcid.org/0000-0003-3747-6611

DOI:

https://doi.org/10.18778/0138-0680.2026.17

Keywords:

superamalgamation property, Fraïssé limit, auxiliary relation, causal space

Abstract

We show that partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser or finer partial order, or a linearization, or an auxiliary relation have the strong amalgamation property, Fraïssé limits and, in many cases, an \(\omega\)-categorical model completion with quantifier elimination.

The same applies to causal spaces, introduced by Kronheimer and Penrose in connection with foundational problems in general relativity. Our main tool is the superamalgamation property, thus we provide arguments suggesting the usefulness of the superamalgamation property also in pure model theory, not only in algebraic logic.

References

G. Bezhanishvili, N. Bezhanishvili, T. Santoli, Y. Venema, A strict implication calculus for compact Hausdorff spaces, Annals of Pure and Applied Logic, vol. 170(11) (2019), pp. 102714, 29, DOI: https://doi.org/10.1016/j.apal.2019.06.003.

G. Bezhanishvili, N. Bezhanishvili, S. Sourabh, Y. Venema, Irreducible equivalence relations, Gleason spaces, and de Vries duality, Applied Categorical Structures. A Journal Devoted to Applications of Categorical Methods in Algebra, Analysis, Computer Science,

Logic, Order and Topology, vol. 25(3) (2017), pp. 381–401, DOI: https://doi.org/10.1007/s10485-016-9434-2.

N. Bezhanishvili, L. Carai, S. Ghilardi, L. Landi, Admissibility of Π2 -inference rules: interpolation, model completion, and contact algebras,Annals of Pure and Applied Logic, vol. 174(1) (2023), pp. Paper No. 103169, 31, DOI: https://doi.org/10.1016/j.apal.2022.103169.

O. Caramello, Theories, sites, toposes, Oxford University Press, Oxford (2018).

N. Caspard, B. Leclerc, B. Monjardet, Finite ordered sets, vol. 144 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2012).

S. Celani, R. Jansana, Bounded distributive lattices with two subordinations, [in:] Mathematics, logic, and their philosophies—essays in honour of Mohammad Ardeshir, vol. 49 of Log. Epistemol. Unity Sci., Springer, Cham (2021), pp. 217–252, DOI: https://doi.org/10.1007/978-3-030-53654-1_8.

S. A. Celani, Amalgamation property in quasi-modal algebras, Revista de la Unión Matemática Argentina, vol. 50(1) (2009), pp. 41–46.

S. A. Celani, Subordination Tarski algebras, Journal of Applied Non-Classical Logics, vol. 29(3) (2019), pp. 288–306, DOI: https://doi.org/10.1080/11663081.2019.1638080.

J. M. Cornejo, H. P. Sankappanavar, Amalgamation Property in the subvarieties of Gautama and Almost Gautama Algebras, Journal of Mahani Mathematical Research, vol. 15(1) (2026), pp. 461–483, DOI: https://doi.org/10.22103/jmmr.2025.26005.1877.

M. Cox, On sets, games and processes (2014), URL: http://etheses.bham.ac.uk/id/eprint/5077, thesis (Ph.D.)–University of Birmingham.

G. D. Dimov, D. Vakarelov, Topological representation of precontact algebras and a connected version of the Stone duality theorem—I,Topology and its Applications, vol. 227 (2017), pp. 64–101, DOI: https://doi.org/10.1016/j.topol.2017.01.019.

I. Düntsch, S. Li, Extension properties of Boolean contact algebras, [in:]Relational and algebraic methods in computer science, vol. 7560 of Lecture Notes in Comput. Sci., Springer, Heidelberg (2012), pp. 342–356, DOI: https://doi.org/10.1007/978-3-642-33314-9_23.

L. Foissy, Plane posets, special posets, and permutations, Advances in Mathematics, vol. 240 (2013), pp. 24–60, DOI: https://doi.org/10.1016/j.aim.2013.03.007.

R. Fraïssé, Theory of relations, revised ed., vol. 145 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam (2000), with an appendix by Norbert Sauer.

D. M. Gabbay, L. Maksimova, Interpolation and definability, vol. 46 of Oxford Logic Guides, The Clarendon Press, Oxford University Press, Oxford (2005), DOI: https://doi.org/10.1093/acprof:oso/9780198511748.001.0001.

S. Ghilardi, A. Gianola, Modularity results for interpolation, amalgamation and superamalgamation, Annals of Pure and Applied Logic, vol. 169(8) (2018), pp. 731–754, DOI: https://doi.org/10.1016/j.apal.2018.04.001.

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, Continuous lattices and domains, vol. 93 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2003), DOI: https://doi.org/10.1017/CBO9780511542725.

G. Grätzer, Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel (2011), DOI: https://doi.org/10.1007/978-3-0348-0018-1.

W. Hodges, Model theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1993), DOI: https://doi.org/10.1017/CBO9780511551574.

B. Jónsson, Universal relational systems, Mathematica Scandinavica, vol. 4 (1956), pp. 193–208, DOI: https://doi.org/10.7146/math.scand.a-10468.

B. Jónsson, Extensions of relational structures, [in:] Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam (1965), pp. 146–157.

I. Kaplan, P. Simon, Automorphism groups of finite topological rank,Transactions of the American Mathematical Society, vol. 372(3) (2019), pp. 2011–2043, DOI: https://doi.org/10.1090/tran/7674.

H. Kihara, H. Ono, Interpolation properties, Beth definability properties and amalgamation properties for substructural logics, Journal of Logic and Computation, vol. 20(4) (2010), pp. 823–875, DOI: https://doi.org/10.1093/logcom/exn084.

E. W. Kiss, L. Márki, P. Pröhle, W. Tholen, Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Scientiarum Mathematicarum Hungarica. Combinatorics, Geometry and Topology (CoGeTo), vol. 18(1) (1982), pp. 79–140.

S. Koppelberg, Handbook of Boolean algebras. Vol. 1, North-Holland Publishing Co., Amsterdam (1989).

C. Kosniowski, A first course in algebraic topology, Cambridge University Press, Cambridge-New York (1980).

E. H. Kronheimer, R. Penrose, On the structure of causal spaces, Proceedings of the Cambridge Philosophical Society, vol. 63 (1967), pp. 481–501, DOI: https://doi.org/10.1017/s030500410004144x.

A. Ledda, H. P. Sankappanavar, G. Vergottini, The Amalgamation Property in the Variety of Regular Double Stone Algebras: A Constructive View, Bulletin of the Section of Logic, vol. 55(1) (2026), pp. 1–47, DOI: https://doi.org/10.18778/0138-0680.2026.01.

P. Lipparini, Comparable binary relations and the amalgamation property, Preprint, arXiv:2301.12482 [math.LO] (2023), URL: https://arxiv.org/abs/2301.12482.

P. Lipparini, A model theory of topology, Studia Logica. An International Journal for Symbolic Logic, vol. 113(1) (2025), pp. 225–259, DOI: https://doi.org/10.1007/s11225-024-10107-3.

P. Lipparini, A hyperamalgamation property, Preprint, arXiv:2606.09329 [math.LO] (2026), URL: https://arxiv.org/abs/2606.09329.

P. Lipparini, Preservation of superamalgamation by expansions, Fundamenta Mathematicae, vol. 272(3) (2026), pp. 205–237, DOI: https://doi.org/10.4064/fm220919-8-11.

J. X. Madarász, Interpolation and amalgamation; pushing the limits. I, Studia Logica. An International Journal for Symbolic Logic, vol. 61(3) (1998), pp. 311–345, DOI: https://doi.org/10.1023/A:1005064504044.

G. Metcalfe, F. Montagna, C. Tsinakis, Amalgamation and interpolation in ordered algebras, Journal of Algebra, vol. 402 (2014), pp. 21–82, DOI: https://doi.org/10.1016/j.jalgebra.2013.11.019.

W. Mielants, Can the best of all possible worlds be a random structure?,Logique et Analyse. Nouvelle Série, vol. 36(141-142) (1993), pp. 61–73.

F. Mynard, E. Pearl (eds.), Beyond topology, vol. 486 of Contemporary Mathematics, American Mathematical Society, Providence, RI (2009), DOI: https://doi.org/10.1090/conm/486.

P. Panangaden, Causality in physics and computation, Theoretical Computer Science, vol. 546 (2014), pp. 10–16, DOI: https://doi.org/10.1016/j.tcs.2014.02.041.

J. S. Russell, On the Probability of Plenitude, Journal of Philosophy, vol. 117(5) (2020), pp. 267–292, DOI: https://doi.org/10.5840/jphil2020117516.

J. H. Schmerl, Countable homogeneous partially ordered sets, Algebra Universalis, vol. 9(3) (1979), pp. 317–321, DOI: https://doi.org/10.1007/BF02488043.