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Birkhoff’s and Mal’cev’s Theorems for Implicational Tonoid Logics


In the context of implicational tonoid logics, this paper investigates analogues of Birkhoff’s two theorems, the so-called subdirect representation and varieties theorems, and of Mal’cev’s quasi-varieties theorem. More precisely, we first recall the class of implicational tonoid logics. Next, we establish the subdirect product representation theorem for those logics and then consider some more related results such as completeness. Thirdly, we consider the varieties theorem for them. Finally, we introduce an analogue of Mal’cev’s quasi-varieties theorem for algebras.

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I dedicate this paper to the late J. Michael Dunn. I would like to thank the reviewer and the Editor in Chief for their helpful comments. This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019S1A5A2A01034874).

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Correspondence to Eunsuk Yang.

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Yang, E. Birkhoff’s and Mal’cev’s Theorems for Implicational Tonoid Logics. Stud Logica (2023).

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  • Implicational tonoid logics
  • Implicational tonoid matrices
  • Subdirect representation
  • Variety
  • Quasi-variety