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Difference–restriction algebras of partial functions: axiomatisations and representations

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Abstract

We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable by injective partial functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists.

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References

  1. Abbott, J.C.: Sets, Lattices, and Boolean Algebras. Allyn and Bacon, Boston (1969)

    MATH  Google Scholar 

  2. Abian, A.: Boolean rings with isomorphisms preserving suprema and infima. J. Lond. Math. Soc. s2-3(4), 618–620 (1971)

  3. Bauer, A., Cvetko-Vah, K., Gehrke, M., van Gool, S.J., Kudryavtseva, G.: A non-commutative Priestley duality. Topol. Appl. 160(12), 1423–1438 (2013)

  4. Berendsen, J., Jansen, D.N., Schmaltz, J., Vaandrager, F.W.: The axiomatization of override and update. J. Appl. Log. 8(1), 141–150 (2010)

    Article  MathSciNet  Google Scholar 

  5. Blackburn, P., Rijke, M.d., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2001)

  6. Borlido, C., McLean, B.: Difference-restriction algebras of partial functions with operators: discrete duality and completion. J. Algebra (in press) (2022). https://doi.org/10.1016/j.jalgebra.2022.03.039

    Article  MATH  Google Scholar 

  7. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symbol. Logic 70(3), 713–740 (2005)

    Article  MathSciNet  Google Scholar 

  8. Egrot, R., Hirsch, R.: Completely representable lattices. Algebra Univ. 67(3), 205–217 (2012)

    Article  MathSciNet  Google Scholar 

  9. Filiot, E., Reynier, P.A.: Transducers, logic and algebra for functions of finite words. ACM SIGLOG News 3(3), 4–19 (2016)

    Article  Google Scholar 

  10. Gehrke, M., Bjarni, J.: Bounded distributive lattices with operators. Math. Japon. 40(2), 207–215 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)

    Article  MathSciNet  Google Scholar 

  12. Gould, V., Hollings, C.: Restriction semigroups and inductive constellations. Commun. Algebra 38(1), 261–287 (2009)

  13. Hirsch, R., Hodkinson, I.: Complete representations in algebraic logic. J. Symbol. Logic 62(3), 816–847 (1997)

    Article  MathSciNet  Google Scholar 

  14. Hirsch, R., Jackson, M., Mikulás, S.: The algebra of functions with antidomain and range. J. Pure Appl. Algebra 220(6), 2214–2239 (2016)

    Article  MathSciNet  Google Scholar 

  15. Hirsch, R., McLean, B.: Disjoint-union partial algebras. Logical Methods Comput. Sci. 13(2:10), 1–31 (2017)

  16. Jackson, M., Stokes, T.: Modal restriction semigroups: towards an algebra of functions. Int. J. Algebra Comput. 21(7), 1053–1095 (2011)

    Article  MathSciNet  Google Scholar 

  17. Jackson, M., Stokes, T.: Monoids with tests and the algebra of possibly non-halting programs. J. Logical Algebraic Methods Program. 84(2), 259–275 (2015)

    Article  MathSciNet  Google Scholar 

  18. Jackson, M., Stokes, T.: Override and update. J. Pure Appl. Algebra 225(3), 106,532 (2021)

  19. Jonsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73(4), 891–939 (1951)

    Article  Google Scholar 

  20. Kudryavtseva, G., Lawson, M.V.: Boolean sets, skew Boolean algebras and a non-commutative Stone duality. Algebra Univ. 75(1), 1–19 (2016)

    Article  MathSciNet  Google Scholar 

  21. Kudryavtseva, G., Lawson, M.V.: A perspective on non-commutative frame theory. Adv. Math. 311, 378–468 (2017)

    Article  MathSciNet  Google Scholar 

  22. Lawson, M.V.: A noncommutative generalization of Stone duality. J. Aust. Math. Soc. 88(3), 385–404 (2010)

    Article  MathSciNet  Google Scholar 

  23. Lawson, M.V.: Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras. Int. J. Algebra Comput. 22(06), 1250, 058 (2012)

  24. Lawson, M.V.: Subgroups of the group of homeomorphisms of the Cantor space and a duality between a class of inverse monoids and a class of Hausdorff étale groupoids. J. Algebra 462, 77–114 (2016)

    Article  MathSciNet  Google Scholar 

  25. Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. Adv. Math. 244, 117–170 (2013)

    Article  MathSciNet  Google Scholar 

  26. Lawson, M.V., Margolis, S.W., Steinberg, B.: The étale groupoid of an inverse semigroup as a groupoid of filters. J. Aust. Math. Soc. 94(2), 234–256 (2013)

    Article  MathSciNet  Google Scholar 

  27. Leech, J.: Normal skew lattices. Semigroup Forum 44(1), 1–8 (1992)

    Article  MathSciNet  Google Scholar 

  28. Leech, J.: Recent developments in the theory of skew lattices. Semigroup Forum 52(1), 7–24 (1996)

    Article  MathSciNet  Google Scholar 

  29. McLean, B.: Complete representation by partial functions for composition, intersection and antidomain. J. Log. Comput. 27(4), 1143–1156 (2017)

    Article  Google Scholar 

  30. McLean, B.: Algebras of partial functions. Ph.D. thesis, University College London (2018)

  31. McLean, B.: A categorical duality for algebras of partial functions. J. Pure Appl. Algebra 225(11), 106 (2021)

  32. Schein, B.M.: Difference semigroups. Comm. Algebra 20(8), 2153–2169 (1992)

    Article  MathSciNet  Google Scholar 

  33. Wagner, V.V.: Generalised groups. Proc. USSR Acad. Sci. 84, 1119–1122 (1952)

    Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for the careful reading of the paper and for their useful suggestions that helped to improve the presentation of our work, in particular allowing us to present the results of Section 4 as a consequence of those from Section 3, thereby making the paper clearer.

Funding

The first author was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES and partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624). The second author was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624) and partially supported by the Research Foundation – Flanders (FWO) under the SNSF–FWO Lead Agency Grant 200021L 196176 (SNSF)/G0E2121N (FWO).

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Authors and Affiliations

  1. CMUC, Departamento de Matemática, Universidade de Coimbra, 3001-501, Coimbra, Portugal

    Célia Borlido

  2. Laboratoire J. A. Dieudonné UMR CNRS 7351, Université Nice Sophia Antipolis, 06108 Nice, Cedex 02, France

    Brett McLean

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  1. Célia Borlido
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  2. Brett McLean
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Correspondence to Célia Borlido.

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Borlido, C., McLean, B. Difference–restriction algebras of partial functions: axiomatisations and representations. Algebra Univers. 83, 24 (2022). https://doi.org/10.1007/s00012-022-00775-4

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