An algebraic analysis of implication in non-distributive logics

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Authors

CHAJDA Ivan EMIR Kadir FAZIO Davide LANGER Helmut LEDDA Antonio PASEKA Jan

Year of publication 2023
Type Article in Periodical
Magazine / Source Journal of logic and computation
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.1093/logcom/exac041
Doi http://dx.doi.org/10.1093/logcom/exac041
Keywords Hilbert algebras; skew Hilbert algebras; pseudocomplemented lattices; sectionally pseudocomplemented lattices; orthomodular lattices; implication algebras
Description In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g. (generalized) orthomodular lattices, and MV-algebras, which admit a natural notion of implication. In fact, it turns out that skew Hilbert algebras play a similar role for (strongly) sectionally pseudocomplemented posets as Hilbert algebras do for relatively pseudocomplemented ones. We will discuss basic properties of closed, dense and weakly dense elements of skew Hilbert algebras and their applications, and we will provide some basic results on their structure theory.
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