The Archimedean property: new horizons and perspectives
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Although there have been repeated attempts to define the concept of an Archimedean algebra for individual classes of residuated lattices, there is no all-purpose definition that suits the general case. We suggest as a possible candidate the notion of a normal-valued and e-cyclic residuated lattice that has the zero radical compact property—namely, a normal-valued and e-cyclic residuated lattice in which every principal convex subuniverse has a trivial radical (understood as the intersection of all its maximal convex subuniverses). We characterize the Archimedean members in the variety of e-cyclic residuated lattices, as well as in various special cases of interest. A theorem to the effect that each Archimedean and prelinear GBL-algebra is commutative, subsuming as corollaries several analogous results from the recent literature, is grist to the mill of our proposal’s adequacy. Finally, we revisit the concept of a hyper-Archimedean residuated lattice, another notion with which researchers have engaged from disparate angles, and investigate some of its properties.
KeywordsResiduated lattice Substructural logic Archimedean property Conrad’s program Lattice-ordered group Generalized BL-algebra Generalized MV-algebra
Mathematics Subject Classification06F05 06D35 06F15 03G10 03B47
Substantive parts of the present paper were written when the first two authors were visiting the Department of Mathematics at Vanderbilt University (Nashville, TN) and when the third author was visiting the Department of Pedagogy, Psychology, Philosophy at the University of Cagliari, Italy. The assistance and facilities provided by both departments are gratefully acknowledged. We also acknowledge the following funding sources that made these visits possible: the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”); the Visiting Scientists Programme of the University of Cagliari, sponsored by Regione Autonoma della Sardegna; the project “Order properties in mathematics and physics”, CUP: F72F16002920002, sponsored by Regione Autonoma della Sardegna; and the Faculty Research Grants Program of the College of Arts and Science of Vanderbilt University. Finally, we thank the anonymous reviewer for his/her careful reading of the manuscript and helpful suggestions.
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