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Difference–restriction algebras of partial functions: axiomatisations and representations


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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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.


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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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).

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  • Partial function
  • Representation
  • Equational axiomatisation
  • Complete representation
  • Atomic

Mathematics Subject Classification

  • 03G10
  • 06A12
  • 06E75