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Densification via polynomials,languages, and frames
Authors:Nikolaos Galatos  Rostislav Horčík
Institution:1. Department of Mathematics, University of Denver, 2390 S. York St., Denver, USA;2. Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Czech Republic;1. Tulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans, LA 70118, USA;2. Department of Mathematics, University of Nebraska – Lincoln, Lincoln, NE 68588, USA;3. University of Education, Hue University, 34 Le Loi St., Hue, Viet Nam;1. Universidad de Cádiz, Puerto Real, Cádiz, Spain;2. CMCC, Universidade Federal do ABC, Santo André, Brazil;3. CMUP, Faculdade de Ciências, Universidade do Porto, Porto, Portugal;4. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia;5. Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, Brazil;6. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal;7. Saint Petersburg University, Saint Petersburg, Russia;1. Utah State University, Department of Mathematics and Statistics, Logan UT 84341, USA;2. Hung Vuong University, Faculty of Natural Sciences, Viet Tri, Phu Tho, Viet Nam;3. Université Bretagne Sud, Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Campus de Tohannic, BP 573 F-56017 Vannes, France
Abstract:It is known that every countable totally ordered set can be embedded into a countable dense one. We extend this result to totally ordered commutative monoids and to totally ordered commutative residuated lattices (the latter result fails in the absence of commutativity). The latter has applications to density elimination of semilinear substructural logics. In particular we obtain as a corollary a purely algebraic proof of the standard completeness of uninorm logic; the advantage over the known proof-theoretic proof and the semantical proof is that it is extremely short and transparent and all details can be verified easily using standard algebraic constructions.
Keywords:Densification  Totally ordered commutative monoid  Residuated lattice
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