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
Abbott, J.C.: Sets, Lattices, and Boolean Algebras. Allyn and Bacon, Boston (1969)
Abian, A.: Boolean rings with isomorphisms preserving suprema and infima. J. Lond. Math. Soc. s2-3(4), 618–620 (1971)
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)
Berendsen, J., Jansen, D.N., Schmaltz, J., Vaandrager, F.W.: The axiomatization of override and update. J. Appl. Log. 8(1), 141–150 (2010)
Blackburn, P., Rijke, M.d., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2001)
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
Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symbol. Logic 70(3), 713–740 (2005)
Egrot, R., Hirsch, R.: Completely representable lattices. Algebra Univ. 67(3), 205–217 (2012)
Filiot, E., Reynier, P.A.: Transducers, logic and algebra for functions of finite words. ACM SIGLOG News 3(3), 4–19 (2016)
Gehrke, M., Bjarni, J.: Bounded distributive lattices with operators. Math. Japon. 40(2), 207–215 (1994)
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)
Gould, V., Hollings, C.: Restriction semigroups and inductive constellations. Commun. Algebra 38(1), 261–287 (2009)
Hirsch, R., Hodkinson, I.: Complete representations in algebraic logic. J. Symbol. Logic 62(3), 816–847 (1997)
Hirsch, R., Jackson, M., Mikulás, S.: The algebra of functions with antidomain and range. J. Pure Appl. Algebra 220(6), 2214–2239 (2016)
Hirsch, R., McLean, B.: Disjoint-union partial algebras. Logical Methods Comput. Sci. 13(2:10), 1–31 (2017)
Jackson, M., Stokes, T.: Modal restriction semigroups: towards an algebra of functions. Int. J. Algebra Comput. 21(7), 1053–1095 (2011)
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)
Jackson, M., Stokes, T.: Override and update. J. Pure Appl. Algebra 225(3), 106,532 (2021)
Jonsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73(4), 891–939 (1951)
Kudryavtseva, G., Lawson, M.V.: Boolean sets, skew Boolean algebras and a non-commutative Stone duality. Algebra Univ. 75(1), 1–19 (2016)
Kudryavtseva, G., Lawson, M.V.: A perspective on non-commutative frame theory. Adv. Math. 311, 378–468 (2017)
Lawson, M.V.: A noncommutative generalization of Stone duality. J. Aust. Math. Soc. 88(3), 385–404 (2010)
Lawson, M.V.: Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras. Int. J. Algebra Comput. 22(06), 1250, 058 (2012)
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)
Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. Adv. Math. 244, 117–170 (2013)
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)
Leech, J.: Normal skew lattices. Semigroup Forum 44(1), 1–8 (1992)
Leech, J.: Recent developments in the theory of skew lattices. Semigroup Forum 52(1), 7–24 (1996)
McLean, B.: Complete representation by partial functions for composition, intersection and antidomain. J. Log. Comput. 27(4), 1143–1156 (2017)
McLean, B.: Algebras of partial functions. Ph.D. thesis, University College London (2018)
McLean, B.: A categorical duality for algebras of partial functions. J. Pure Appl. Algebra 225(11), 106 (2021)
Schein, B.M.: Difference semigroups. Comm. Algebra 20(8), 2153–2169 (1992)
Wagner, V.V.: Generalised groups. Proc. USSR Acad. Sci. 84, 1119–1122 (1952)
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.
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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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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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DOI: https://doi.org/10.1007/s00012-022-00775-4