A variety of algebras closely related to subordination algebras

Abstract

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

Keywords:

• Subordination algebras
• precontact algebras
• region-based theory of space
• modal logic

2010 Mathematics Subject Classifications:

• 03B45
• 03G05
• 03G25
• 06E15
• 06E75

Acknowledgments

We thank the reviewers of the paper for their helpful comments that have contributed significantly to the final shape of the paper and to improve its readability and content.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We have some objections to using the label ‘strict implication’ because in modal algebras the strict implication is usually the operation defined by the term $◻(x\to y)$. This operation, call it $⊸$, satisfies $(x⊸y)\wedge (y⊸z)\le x⊸z$, a property that does not necessarily hold for the characteristic function of a subordination relation because these relations do not need to be transitive.

2 We will use the term ‘open’ applied to filters in a different sense. On pseudo-contact algebras our open filters and the modal filters will coincide, but not on pseudo-subordination algebras.

3 Our definition of the ternary relation follows Blackburn et al. (2001) and departs from Goldblatt Goldblatt (1989) in that the first component of the triple $〈x,y,z〉$ is in Goldblatt (1989) the third one.

4 The original, equivalent condition, given by Halmos is: for every clopen set U of $X_2$, $R^{-1}\left[U\right]$ is a clopen set of $X_1$. In Bezhanishvili, Bezhanishvili, Sourabh, et al. (2017) the Boolean relations on a Stone space are called Esakia relations.

5 We hope that the use of T as a symbol of the formal language ${\mathcal{L}}_T$ and as a mathematical symbol to denote ternary relations will not cause any misunderstanding.

Additional information

Funding

This project has received funding from the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 689176. The first author was also partially supported by the research project PICT 2019-00882 (2021–2023) of the Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación (Argentina). And the second author was also partially supported by the research grant 2014 SGR 788 from the government of Catalonia and by the research projects MTM2011-25747 and MTM2016-74892-P from the Spanish Ministerio de Economía y Competitividad, which include feder funds from the European Union.