On the logic of distributive nearlattices

• Published in: Mathematical logic quarterly Vol. 68; no. 3; pp. 375 - 385
• Main Author: González, Luciano J.
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.08.2022
• Subjects: Logic
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202200012

We study the propositional logic SDN$\mathcal {S}_\mathbb {DN}$ associated with the variety of distributive nearlattices DN$\mathbb {DN}$. We prove that the logic SDN$\mathcal {S}_\mathbb {DN}$ coincides with the assertional logic associated with the variety DN$\mathbb {DN}$ and with the order‐based logic associated with DN$\mathbb {DN}$. We obtain a characterization of the reduced matrix models of logic SDN$\mathcal {S}_\mathbb {DN}$. We develop a connection between the logic SDN$\mathcal {S}_\mathbb {DN}$ and the {∧,∨,⊤}$\lbrace \wedge ,\vee ,\top \rbrace$‐fragment of classical logic. Finally, we present two Hilbert‐style axiomatizations for the logic SDN$\mathcal {S}_\mathbb {DN}$.