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Difference–restriction algebras of partial functions : axiomatisations and representations

Célia Borlido and Brett McLean (UGent)
(2022) ALGEBRA UNIVERSALIS. 83(3).
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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.
Keywords
Algebra and Number Theory, Partial function, Representation, Equational axiomatisation, Complete representation, Atomic, SEMIGROUPS

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Citation

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MLA
Borlido, Célia, and Brett McLean. “Difference–Restriction Algebras of Partial Functions : Axiomatisations and Representations.” ALGEBRA UNIVERSALIS, vol. 83, no. 3, 2022, doi:10.1007/s00012-022-00775-4.
APA
Borlido, C., & McLean, B. (2022). Difference–restriction algebras of partial functions : axiomatisations and representations. ALGEBRA UNIVERSALIS, 83(3). https://doi.org/10.1007/s00012-022-00775-4
Chicago author-date
Borlido, Célia, and Brett McLean. 2022. “Difference–Restriction Algebras of Partial Functions : Axiomatisations and Representations.” ALGEBRA UNIVERSALIS 83 (3). https://doi.org/10.1007/s00012-022-00775-4.
Chicago author-date (all authors)
Borlido, Célia, and Brett McLean. 2022. “Difference–Restriction Algebras of Partial Functions : Axiomatisations and Representations.” ALGEBRA UNIVERSALIS 83 (3). doi:10.1007/s00012-022-00775-4.
Vancouver
1.
Borlido C, McLean B. Difference–restriction algebras of partial functions : axiomatisations and representations. ALGEBRA UNIVERSALIS. 2022;83(3).
IEEE
[1]
C. Borlido and B. McLean, “Difference–restriction algebras of partial functions : axiomatisations and representations,” ALGEBRA UNIVERSALIS, vol. 83, no. 3, 2022.
@article{01GK6SK87YAZ6CY7ZH323N1NYH,
  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.}},
  articleno    = {{24}},
  author       = {{Borlido, Célia and McLean, Brett}},
  issn         = {{0002-5240}},
  journal      = {{ALGEBRA UNIVERSALIS}},
  keywords     = {{Algebra and Number Theory,Partial function,Representation,Equational axiomatisation,Complete representation,Atomic,SEMIGROUPS}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{27}},
  title        = {{Difference–restriction algebras of partial functions : axiomatisations and representations}},
  url          = {{http://doi.org/10.1007/s00012-022-00775-4}},
  volume       = {{83}},
  year         = {{2022}},
}
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