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On Hilbert algebras generated by the order

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Abstract

In this paper we study the variety of order Hilbert algebras, which is the equivalent algebraic semantics of the order implicational calculus of Bull (J Symb Logic, 29:33–34, 1964).

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Notes

  1. Let \((X,\le )\) be a poset and \(x,y\in X\). It holds that \(x\le y\) whenever \(\alpha (x)\le \alpha (y)\). Indeed, let \(\alpha (x) \le \alpha (y)\). Let \(f:X \rightarrow \mathrm {D}(X)\) be the morphism of posets given by \(f(z) = (z]\), where \(\mathrm {D}(X)\) is the completely distributive lattice of downsets of \((X,\le )\). Thus, there exists an unique morphism of complete lattices \(h:{\mathsf {C}}_p(X) \rightarrow \mathrm {D}(X)\) such that \(f = h \circ \alpha \). Since \(\alpha (x) \le \alpha (y)\) then \((h\circ \alpha )(x) \le (h\circ \alpha )(y)\), i.e., \((x]\subseteq (y]\). Hence, \(x\le y\), which was our aim. In particular, \(\alpha \) is an injective map.

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Funding

Funding was provided by Universidad Nacional de La Plata (Grant Number PID X921).

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Correspondence to J. L. Castiglioni.

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Castiglioni, J.L., Celani, S.A. & San Martín, H.J. On Hilbert algebras generated by the order. Arch. Math. Logic 61, 155–172 (2022). https://doi.org/10.1007/s00153-021-00777-4

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