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RECURSIVE AXIOMATISATIONS FROM SEPARATION PROPERTIES

Part of: General logic Algebraic structures Model theory Graph theory

Published online by Cambridge University Press:  03 February 2021

ROB EGROT
ROB EGROT*
Affiliation:
FACULTY OF INFORMATION AND COMMUNICATION TECHNOLOGY MAHIDOL UNIVERSITY 999 PHUTTHAMONTHON 4 RD, SALAYA NAKHON PATHOM 73170, THAILANDE-mail: robert.egr@mahidol.ac.th
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Abstract

We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.

Keywords

first-order axiomatisabilitygenerating recursive axiomatisationsrepresentable posetsgraph colouringsharmonious colouringsdisjoint union partial algebras

MSC classification

Primary: 03C98: Applications of model theory
Secondary: 03B70: Logic in computer science 05C15: Coloring of graphs and hypergraphs 08A55: Partial algebras
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1228 - 1258
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Copyright
© Association for Symbolic Logic 2021

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References

REFERENCES

Abian, A., Boolean rings with isomorphisms preserving suprema and infima . Journal of the London Mathematical Society (2) , vol. 3 (1971), pp. 618–620.10.1112/jlms/s2-3.4.618CrossRefGoogle Scholar
Balbes, R., A representation theory for prime and implicative semilattices . Transactions of the American Mathematical Society , vol. 136 (1969), pp. 261–267.10.1090/S0002-9947-1969-0233741-7CrossRefGoogle Scholar
Birkhoff, G., On the combination of subalgebras . Proceedings of the Cambridge Philosophical Society , vol. 29, (1933), pp. 441–464.10.1017/S0305004100011464CrossRefGoogle Scholar
Birkhoff, G. and Frink, O. Jr., Representations of lattices by sets . Transactions of the American Mathematical Society , vol. 64 (1948), pp. 299–316.10.1090/S0002-9947-1948-0027263-2CrossRefGoogle Scholar
Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York and Berlin, 1981.10.1007/978-1-4613-8130-3CrossRefGoogle Scholar
Chang, C. C., On the representation of $\alpha$ -complete Boolean algebras. Transactions of the American Mathematical Society, vol. 85 (1957), pp. 208–218.Google Scholar
Chang, C. C. and Horn, A., On the representation of $\alpha$ -complete lattices. Fundamenta Mathematicae, vol. 51 (1962/1963), pp. 253–258.10.4064/fm-51-3-253-258CrossRefGoogle Scholar
Cheng, Y. and Kemp, P., Representation of posets. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 38 (1992), pp. 269–276.10.1002/malq.19920380122CrossRefGoogle Scholar
Cornish, W. H. and Hickman, R. C., Weakly distributive semilattices . Acta Mathematica Academiae Scientiarum Hungaricae , vol. 32 (1978), no. 1–2, pp. 5–16.CrossRefGoogle Scholar
de Bruijn, N. G. and Erdös, P., A colour problem for infinite graphs and a problem in the theory of relations . Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A. 54 = Indagationes Mathematicae , vol. 13 (1951), pp. 369–373.Google Scholar
Egrot, R., Representable posets . Journal of Applied Microbiology , vol. 16 (2016), pp. 60–71.Google Scholar
Egrot, R., Closure operators, frames and neatest representations . Bulletin of the Australian Mathematical Society , vol. 96 (2017), no. 3, pp. 361–373.CrossRefGoogle Scholar
Egrot, R., Non-elementary classes of representable posets. Proceedings of the American Mathematical Society , vol. 145 (2017), no. 11, pp. 4675–4685.10.1090/proc/13636CrossRefGoogle Scholar
Egrot, R., No finite axiomatizations for posets embeddable into distributive lattices . Annals of Pure and Applied Logic , vol. 169 (2018), no. 3, pp. 235–242.CrossRefGoogle Scholar
Egrot, R., Recursive axiomatizations for representable posets . International Journal of Algebra and Computation, vol. 29 (2019), no. 4, pp. 699–711.10.1142/S021819671950022XCrossRefGoogle Scholar
Egrot, R. and Hirsch, R., Completely representable lattices. Algebra Universalis, vol. 67 (2012), pp. 205–217.10.1007/s00012-012-0181-4CrossRefGoogle Scholar
Erdős, P., Graph theory and probability. Canadian Journal of Mathematics, vol. 11 (1959), pp. 34–38.10.4153/CJM-1959-003-9CrossRefGoogle Scholar
Fagin, R., Contributions to the model-theory of finite-structures (ProQuest LLC, Ann Arbor, MI), Ph.D. thesis, University of California, Berkeley, 1973.Google Scholar
Frank, O., Harary, F., and Plantholt, M., The line-distinguishing chromatic number of a graph. Ars Combinatoria, vol. 14 (1982), pp. 241–252.Google Scholar
Frayne, T., Morel, A. C., and Scott, D. S., Reduced direct products . Fundamenta Mathematicae , vol. 51 (1962/1963), pp. 195–228.CrossRefGoogle Scholar
Hickman, R. C. and Monro, G. P., Distributive partially ordered sets . Fundamenta Mathematicae , vol. 120 (1984), no. 2, pp. 151–166.10.4064/fm-120-2-151-166CrossRefGoogle Scholar
Hirsch, R. and Hodkinson, I., Representability is not decidable for finite relation algebras. Transactions of the American Mathematical Society, vol. 353 (2001), no. 4, pp. 1403–1425.CrossRefGoogle Scholar
Hirsch, R. and Hodkinson, I., Relation Algebras by Games , North-Holland, Amsterdam, 2002.Google Scholar
Hirsch, R. and McLean, B., Disjoint-union partial algebras . Logical Methods in Computer Science , vol. 13 (2017), no. 2, Article no. 10, pp. 1–31.Google Scholar
Hodges, W., Model Theory , Encyclopedia of Mathematics and Its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.10.1017/CBO9780511551574CrossRefGoogle Scholar
Hopcroft, J. E. and Krishnamoorthy, M. S., On the harmonious coloring of graphs . SIAM Journal on Algebraic and Discrete Methods , vol. 4 (1983), no. 3, pp. 306–311.10.1137/0604032CrossRefGoogle Scholar
Kearnes, K. A., The class of prime semilattices is not finitely axiomatizable. Semigroup Forum, vol. 55 (1997), no. 1, pp. 133–134.10.1007/PL00005908CrossRefGoogle Scholar
Keisler, H., Ultraproducts and elementary models . Indagationes Mathematicae , vol. 23 (1961), pp. 477–495.10.1016/S1385-7258(61)50048-0CrossRefGoogle Scholar
Kemp, P., Representation of partially ordered sets . Algebra Universalis , vol. 30 (1993), pp. 348–351.10.1007/BF01190444CrossRefGoogle Scholar
König, D., Sur les correspondances multivoques des ensembles . Fundamenta Mathematicae , vol. 8, (1926), pp. 114–134.10.4064/fm-8-1-114-134CrossRefGoogle Scholar
Loomis, L., On the representation of $\sigma$ -complete Boolean algebras . Bulletin of the American Mathematical Society , vol. 53 (1947), pp. 757–760.CrossRefGoogle Scholar
Łoś, J., Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, Mathematical Interpretation of Formal Systems, North-Holland, Amsterdam, 1955, pp. 98–113.CrossRefGoogle Scholar
Schein, B. M., On the definition of distributive semilattices . Algebra Universalis , vol. 2 (1972), pp. 1–2.10.1007/BF02945000CrossRefGoogle Scholar
Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers . Israel Journal of Mathematics , vol. 10 (1971), pp. 224–233.10.1007/BF02771574CrossRefGoogle Scholar
Sikorski, R., On the representation of Boolean algebras as fields of sets. Fundamenta Mathematicae, vol. 35 (1948), pp. 247–258.10.4064/fm-35-1-247-258CrossRefGoogle Scholar
Stone, M., The theory of representations for Boolean algebras . Transactions of the American Mathematical Society , vol. 40 (1936), pp. 37–111.Google Scholar
Tarski, A., Contributions to the theory of models, III . Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen , vol. 58 (1955), pp. 56–64.Google Scholar
Taylor, W., Atomic compactness and graph theory . Fundamenta Mathematicae , vol. 65 (1969), pp. 139–145.10.4064/fm-65-2-139-145CrossRefGoogle Scholar
Van Alten, C. J., Embedding ordered sets into distributive lattices . Order , vol. 33 (2016), no. 3, pp. 419–427.10.1007/s11083-015-9376-6CrossRefGoogle Scholar
Vardi, M., On the complexity of bounded-variable queries. ACM Symposium on Principles of Database Systems , ACM Press, New York, May 1995, pp. 266–276.Google Scholar
Wheeler, W. H., The first order theory of $N$ -colorable graphs. Transactions of the American Mathematical Society, vol. 250 (1979), pp. 289–310.Google Scholar