×

zbMATH — the first resource for mathematics

Admissibility via natural dualities. (English) Zbl 1346.08005
Summary: It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.

MSC:
08C15 Quasivarieties
08C20 Natural dualities for classes of algebras
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
08B20 Free algebras
06D50 Lattices and duality
03C05 Equational classes, universal algebra in model theory
03C13 Model theory of finite structures
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Lorenzen, P., Einführung in die operative logik und Mathematik, Grundlehren Math. Wiss., vol. 78, (1955), Springer · Zbl 0066.24802
[2] Gentzen, G., Untersuchungen über das logische schliessen, Math. Z., 39, 176-210, (1935), 405-431 · Zbl 0010.14501
[3] Johansson, I., Der minimalkalkul, ein reduzierter intuitionistischer formalismus, Compos. Math., 4, 119-136, (1936) · JFM 62.1045.08
[4] Whitman, P., Free lattices, Ann. Math., 42, 325-329, (1941) · JFM 67.0085.01
[5] Metcalfe, G.; Olivetti, N.; Gabbay, D., Proof theory for fuzzy logics, (2008), Springer
[6] Metcalfe, G.; Montagna, F., Substructural fuzzy logics, J. Symb. Log., 72, 834-864, (2007) · Zbl 1139.03017
[7] Rybakov, V., A criterion for admissibility of rules in the modal system S4 and the intuitionistic logic, Algebra Log., 23, 369-384, (1984) · Zbl 0598.03013
[8] Rybakov, V., Admissibility of logical inference rules, Stud. Logic Found. Math., vol. 136, (1997), Elsevier Amsterdam · Zbl 0872.03002
[9] Iemhoff, R., On the admissible rules of intuitionistic propositional logic, J. Symb. Log., 66, 281-294, (2001) · Zbl 0986.03013
[10] Rozière, P., Regles admissibles en calcul propositionnel intuitionniste, (1992), Université Paris VII, Ph.D. thesis
[11] Iemhoff, R., Intermediate logics and Visser’s rules, Notre Dame J. Form. Log., 46, 65-81, (2005) · Zbl 1102.03032
[12] Cintula, P.; Metcalfe, G., Admissible rules in the implication-negation fragment of intuitionistic logic, Ann. Pure Appl. Log., 162, 162-171, (2010) · Zbl 1225.03011
[13] Jeřábek, E., Admissible rules of modal logics, J. Log. Comput., 15, 411-431, (2005) · Zbl 1077.03011
[14] Babenyshev, S.; Rybakov, V., Unification in linear temporal logic LTL, Ann. Pure Appl. Log., 162, 991-1000, (2011) · Zbl 1241.03014
[15] Babenyshev, S.; Rybakov, V., Linear temporal logic LTL: basis for admissible rules, J. Log. Comput., 21, 157-177, (2011) · Zbl 1233.03026
[16] Jeřábek, E., Admissible rules of łukasiewicz logic, J. Log. Comput., 20, 425-447, (2010) · Zbl 1216.03042
[17] Jeřábek, E., Bases of admissible rules of łukasiewicz logic, J. Log. Comput., 20, 1149-1163, (2010) · Zbl 1216.03043
[18] Ghilardi, S., Unification in intuitionistic logic, J. Symb. Log., 64, 859-880, (1999) · Zbl 0930.03009
[19] Ghilardi, S., Best solving modal equations, Ann. Pure Appl. Log., 102, 184-198, (2000) · Zbl 0949.03010
[20] Clark, D. M.; Davey, B. A., Natural dualities for the working algebraist, (1998), Cambridge University Press · Zbl 0910.08001
[21] Metcalfe, G.; Röthlisberger, C., Admissibility in De Morgan algebras, Soft Comput., 16, 1875-1882, (2012) · Zbl 1281.06009
[22] Metcalfe, G.; Röthlisberger, C., Admissibility in finitely generated quasivarieties, Log. Methods Comput. Sci., 9, 1-19, (2013) · Zbl 1297.03009
[23] Belnap, N., How a computer should think, (Ryle, G., Contemporary Aspects of Philosophy, (1977), Oriel Press Ltd.), 30-56
[24] Gehrke, M.; Walker, C.; Walker, E., Fuzzy logics arising from strict De Morgan systems, (Topological and Algebraic Structures in Fuzzy Sets, Trends Log., vol. 20, (2003)), 257-276 · Zbl 1055.03015
[25] Lane, S. M., Categories for the working Mathematician, Grad. Texts Math., vol. 5, (1971), Springer · Zbl 0232.18001
[26] Burris, S.; Sankappanavar, H. P., A course in universal algebra, Grad. Texts Math., vol. 78, (1981), Springer-Verlag New York · Zbl 0478.08001
[27] Chang, C.; Keisler, H., Model theory, Stud. Logic Found. Math., vol. 73, (1977), Elsevier
[28] Jónsson, B., Sublattices of a free lattice, Can. J. Math., 13, 256-264, (1961) · Zbl 0132.26201
[29] Fuchs, L., Abelian groups, (1960), Pergamon Press · Zbl 0100.02803
[30] Bergman, C., Structural completeness in algebra and logic, (Andréka, H.; Monk, J.; Nemeti, I., Algebraic Logic, Colloq. Math. Soc. János Bolyai, vol. 54, (1991), North-Holland Amsterdam), 59-73 · Zbl 0749.08007
[31] Beynon, W. M., Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Can. J. Math., 29, 243-254, (1977) · Zbl 0361.06017
[32] Priestley, H. A., Representation of distributive lattices by means of ordered stone spaces, Bull. Lond. Math. Soc., 2, 186-190, (1970) · Zbl 0201.01802
[33] Balbes, R.; Horn, A., Projective distributive lattices, Pac. J. Math., 33, 273-279, (1970) · Zbl 0185.03602
[34] Grätzer, G., General lattice theory, (1998), Birkhäuser Basel · Zbl 0385.06015
[35] Davey, B. A., Dualities for stone algebras, double stone algebras and relative stone algebras, Colloq. Math. Soc. János Bolyai, 46, 1-14, (1982) · Zbl 0514.06008
[36] Pynko, A. P., Implicational classes of De Morgan lattices, Discrete Math., 205, 171-181, (1999) · Zbl 0938.06009
[37] Kalman, J. A., Lattices with involution, Trans. Am. Math. Soc., 87, 485-491, (1958) · Zbl 0228.06003
[38] Bova, S.; Cabrer, L. M., Unification and projectivity in De Morgan and Kleene algebras, Order, 31, 159-187, (2014) · Zbl 1303.06009
[39] Davey, B. A.; Werner, H., Piggyback-dualities, Bull. Aust. Math. Soc., 32, 1-32, (1985) · Zbl 0609.08004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.