In this notes, we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with \(\mathbb Q\)-vector lattices, we present the divisible hull as a categorical adjunction, and we prove a duality between finitely presented algebras and rational polyhedra.
KeywordsDMV-algebras MV-algebras Rational Łukasiewicz logic Divisible hull Rational polyhedra
S. Lapenta acknowledges partial support from the Italian National Research Project (PRIN2010-11) entitled Metodi logici per il trattamento dellinformazione. I. Leuştean was supported by a Grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0730.
Compliance with ethical standards
Conflict of interest
Both authors declare that they had no conflict of interest in writing this paper.
This article does not contain any studies with human participants or animals performed by any of the authors.
- Birkoff G (1967) Lattice theory, 3rd edn. AMS Colloquium Publications, New yorkGoogle Scholar
- Di Nola A, Leuştean I (2011) Łukasiewicz Logic and MV-algebras. In: Cintula P et al (eds) Handbook of Mathematical Fuzzy Logic. Studies in Logic. College Publications, LondonGoogle Scholar
- Di Nola A, Lenzi G, Vitale G (2016) Riesz-McNaughton functions and Riesz MV-algebras of nonlinear functions. Fuzzy Sets and Systems. doi: 10.1016/j.fss.2016.03.003
- Gerla B (2001) Rational Łukasiewicz logic and divisible MV-algebras. Neural Netw World 10(11):579–584Google Scholar
- Lapenta S (2015) MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Łukasiewicz logic, PhD ThesisGoogle Scholar
- Lapenta S, Leuştean I (2015) Towards understanding the Pierce-Birkhoff conjecture via MV-algebras. Fuzzy Sets Syst 276:114–130Google Scholar
- Lapenta S, Leuştean I (2016) Scalar extensions for the algebraic structures of Łukasiewicz logic. J Pure Appl Algebra 220:1538–1553Google Scholar
- Madden J (1985) \(l\)-groups of piecewise linear functions, Ordered algebraic structures (Cincinnati, Ohio, 1982), Lecture Notes in Pure and Appl Math 99:117–124Google Scholar
- Mundici D (2011) Advances in Łukasiewicz calculus and MV-algebras, Trends in Logic, vol 35. SpringerGoogle Scholar