Abstract
Motivated by Gentzen’s disjunction elimination rule in his Natural Deduction calculus and reading inequalities with meet in a natural way, we conceive a notion of distributivity for join semilattices. We prove that it is equivalent to a notion present in the literature. In the way, we prove that all notions of distributivity for join semilattices we have found in the literature are linearly ordered. We finally consider the notion of distributivity in join semilattices with arrow, that is, the algebraic structure corresponding to the disjunction-conditional fragment of intuitionistic logic.
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Notes
- 1.
A non-empty set I of a meet semillatice is called strong ideal if for any finite subset \(S \subseteq I\), it holds that \(S^{ul} \subseteq I\).
- 2.
Note that the original Hickman’s statement can be misleading since the condition “there exists \((x \wedge a_1) \vee (x \wedge a_2) \vee \cdots \vee (x \wedge a_n)\)” is missing.
- 3.
We thank the author of this PhD thesis for communicating this example.
- 4.
We thank the referee for pointing out this short proof.
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Acknowledgements
The authors thank the anonymous referee for his/her helpful comments. They also acknowledge partial support by the H2020 MSCA-RISE-2015 project SYSMICS. Esteva and Godo also acknowledge the FEDER/MINECO project RASO (TIN2015-71799-C2-1-P).
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Ertola-Biraben, R.C., Esteva, F., Godo, L. (2021). On Distributive Join Semilattices. In: Fazio, D., Ledda, A., Paoli, F. (eds) Algebraic Perspectives on Substructural Logics. Trends in Logic, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-030-52163-9_3
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