The category of \(L\)-algebras. (English) Zbl 1536.18003
Summary: The category \(\mathbf{LAlg}\) of \(L\)-algebras is shown to be complete and cocomplete, regular with a zero object and a projective generator, normal and subtractive, ideal determined, but not Barr-exact. Originating from algebraic logic, \(L\)-algebras arise in the theory of Garside groups, measure theory, functional analysis, and operator theory. It is shown that the category \(\mathbf{LAlg}\) is far from protomodular, but it has natural semidirect products which have not been described in category-theoretic terms.
MSC:
| 18C05 | Equational categories |
| 08C05 | Categories of algebras |
| 18D30 | Fibered categories |
| 06F05 | Ordered semigroups and monoids |
| 08A55 | Partial algebras |
| 18B10 | Categories of spans/cospans, relations, or partial maps |
| 18C10 | Theories (e.g., algebraic theories), structure, and semantics |
References:
| [1] | J. Adámek, J. Rosický: Locally presentable and accessible categories, Cambridge University Press, 1994 · Zbl 0795.18007 |
| [2] | E. Artin: Theory of braids, Ann. of Math. 48 (1947), 101-126 · Zbl 0030.17703 |
| [3] | M. Barr: Exact categories, in: Exact Categories and Sheaves, Lecture Notes in Math. 236, Springer, New York (1971), l-120 · Zbl 0223.18009 |
| [4] | J. Bénabou: Introduction to bicategories, Lecture Notes in Mathematics 47 (Springer, Berlin, 1967), 1-77 · Zbl 1375.18001 |
| [5] | J. Bénabou, J. Roubaud: Monades et descente, C. R. Acad. Sci. 270 (1970), 96-98 · Zbl 0287.18007 |
| [6] | A. Bigard, K. Keimel, and S. Wolfenstein: Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 · Zbl 0384.06022 |
| [7] | W. J. Blok, D. Pigozzi: Algebraizable logics, Mem. Amer. Math. Soc. 336 (1989), 85pp. · Zbl 0664.03042 |
| [8] | D. Bourn: Normalization equivalence, kernel equivalence and affine categories, Springer Lecture Notes Math. 1488 (1991), 43-62 · Zbl 0756.18007 |
| [9] | D. Bourn: Mal’cev categories and fibrations of pointed objects, Appl. Categ. Struct. 4 (1996), 307-327 · Zbl 0856.18004 |
| [10] | D. Bourn: Protomodular aspect of the dual of a topos, Adv. Math. 187 (2004), 240-255 · Zbl 1047.18006 |
| [11] | D. Bourn, G. Janelidze: Protomodularity, descent, and semidirect products, Theor. Appl. Cat 4 (1998), 37-46 · Zbl 0890.18003 |
| [12] | D. Bourn, G. Janelidze: Characterization of protomodular varieties of universal algebras, Theor. Appl. Categories 11 (2003), 143-147 · Zbl 1021.08003 |
| [13] | D. Bourn, N. Martins-Ferreira, A. Montoli, M. Sobral: Monoids and pointed S-protomodular categories, Homology, Homotopy, Appl. 18 (2016), 151-172 · Zbl 1350.18019 |
| [14] | E. Brieskorn, K. Saito: Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271 · Zbl 0243.20037 |
| [15] | T. Bühler: Exact categories, Expo. Math. 28 (2010), no. 1, 1-69 · Zbl 1192.18007 |
| [16] | F. Buekenhout, E. Shult: On the foundations of polar geometry, Geometriae Dedi-cata 3 (1974), 155-170 · Zbl 0286.50004 |
| [17] | A. Carboni, J. Lambek, M. C. Pedicchio: Diagram chasing in Mal’cev categories, J. Pure Appl. Algebra 69 (1991), 271-284 · Zbl 0722.18005 |
| [18] | A. Carboni, E. M. Vitale: Regular and exact completions, J. Pure Appl. Algebra 125 (1998), 79-116 · Zbl 0891.18002 |
| [19] | A. Carboni, R. F. C. Walters: Cartesian bicategories, I, J. Pure Appl. Algebra 49 (1987), 11-32 · Zbl 0637.18003 |
| [20] | C. C. Chang: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467-490 · Zbl 0084.00704 |
| [21] | C. C. Chang: A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74-80 · Zbl 0093.01104 |
| [22] | P. M. Cohn: Universal Algebra, Dordrecht, Netherlands: D.Reidel Publishing, 1981 · Zbl 0461.08001 |
| [23] | M. R. Darnel: Theory of lattice-ordered groups, Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York, 1995 · Zbl 0810.06016 |
| [24] | P. Dehornoy: Groupes de Garside, Ann. Sci.École Norm. Sup. (4) 35 (2002), no. 2, 267-306 · Zbl 1017.20031 |
| [25] | P. Dehornoy, L. Paris: Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604 · Zbl 1030.20021 |
| [26] | P. Dehornoy, F. Digne, E. Godelle, D. Krammer, J. Michel: Foundations of Garside Theory, EMS Tracts in Math. 22, European Mathematical Society, 2015 · Zbl 1370.20001 |
| [27] | P. Deligne: Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302 · Zbl 0238.20034 |
| [28] | A. Dvurečenskij: Pseudo MV-algebras are intervals in l-groups, J. Austral. Math. Soc. 72 (2002), 427-445 · Zbl 1027.06014 |
| [29] | A. Dvurečenskij, S. Pulmannová: New trends in quantum structures, Mathematics and its Applications, 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000 · Zbl 0987.81005 |
| [30] | P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209 · Zbl 0969.81030 |
| [31] | Tomas Everaert: Effective descent morphisms of regular epimorphisms, J. Pure Appl. Algebra 216 (2012), 1896-1904 · Zbl 1257.18001 |
| [32] | D. H. Fremlin: Measure theory, Vol. 3, Measure algebras, Corrected second printing of the 2002 original, Torres Fremlin, Colchester, 2004 · Zbl 1165.28002 |
| [33] | P. Freyd, A. Scedrov: Categories, Allegories, North-Holland, Amsterdam, 1990 · Zbl 0698.18002 |
| [34] | F. A. Garside: The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969) 235-254 · Zbl 0194.03303 |
| [35] | J. Gispert, D. Mundici: MV-algebras: a variety for magnitudes with Archimedean units, Algebra univers. 53 (2005), 7-43 · Zbl 1093.06010 |
| [36] | P. A. Grillet: Regular categories, in: Exact Categories and Categories of Sheaves, Lecture Notes in Math. 236, Springer Verlag, Berlin-Heidelberg-NewYork, 1971 · Zbl 0223.18009 |
| [37] | M. Gran, Z. Janelidze, A. Ursini: A good theory of ideals in regular multi-pointed categories, J. Pure Appl. Algebra 216 (2012), 1905-1919 · Zbl 1257.18011 |
| [38] | H. P. Gumm, A. Ursini: Ideals in universal algebras, Algebra Univ. 19 (1984), 45-54 · Zbl 0547.08001 |
| [39] | S. S. Holland, Jr.: Orthomodularity in infinite dimensions; a theorem of M. Solèr, Bull. Amer. Math. Soc. 32 (1995), no. 2, 205-234 · Zbl 0856.11021 |
| [40] | Z. Janelidze: Subtractive categories, Appl. Categ. Struct. 13 (2005), 343-350 · Zbl 1095.18002 |
| [41] | Z. Janelidze: The pointed subobject functor, 3 ¢ 3 lemmas, and subtractivity of spans, Theor. Appl. Categories 23 (2010), 221-242 · Zbl 1234.18009 |
| [42] | G. Janelidze, W. Tholen: Facets of descent II, Applied Categ. Struct. 5 (1997), 229-248 · Zbl 0880.18007 |
| [43] | G. Janelidze, L. Márki, W. Tholen: Semi-abelian categories, J. Pure Appl. Alge-bra 168 (2002), 367-386 · Zbl 0993.18008 |
| [44] | G. Janelidze, L. Márki, A. Ursini: Ideals and clots in pointed regular categories, Appl. Categ. Struct. 17 (2009), 345-350 · Zbl 1183.18002 |
| [45] | G. Janelidze, L. Márki, W. Tholen, A. Ursini: Ideal determined categories, Cahiers de Topol. Géom. Différentielle Catég. 51 (2010), 115-125 · Zbl 1208.18001 |
| [46] | G. Kalmbach: Orthomodular lattices, London Mathematical Society Monographs, 18, Academic Press, Inc., London, 1983 · Zbl 0512.06011 |
| [47] | B. Keller: Chain complexes and stable categories, Manuscr. math. 67 (1990), 379-417 · Zbl 0753.18005 |
| [48] | G. M. Kelly: Monomorphisms, Epimorphisms, and Pull-Backs, J. Austral. Math. Soc. 9 (1969), 124-142 · Zbl 0169.32604 |
| [49] | P. Köhler: The semigroup of varieties of Brouwerian semilattices, Trans. Amer. Math. Soc. 241 (1978), 331-342 · Zbl 0389.06008 |
| [50] | F. W. Lawvere: Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA 50 (1963), 869-872 · Zbl 0119.25901 |
| [51] | F. W. Lawvere: Theory of Categories over a Base Topos, Ist. Mat. Univ. Perugia (1972-73) |
| [52] | W. A. J. Luxemburg, A. C. Zaanen: Riesz spaces, Vol. I., North-Holland Publishing Co., Amsterdam-London; · Zbl 0231.46014 |
| [53] | American Elsevier Publishing Co., New York, 1971 |
| [54] | S. Mac Lane: Duality for groups, Bull. Amer. Math. Soc. 56 (1950), 485-516 · Zbl 0041.36306 |
| [55] | S. Mac Lane, I. Moerdijk: Sheaves in geometry and logic, A first introduction to topos theory, Corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994 |
| [56] | D. Maharam: The representation of abstract measure functions, Trans. Amer. Math. Soc. 65 (1949), 279-330 · Zbl 0036.31401 |
| [57] | N. Martins-Ferreira, M. Sobral: On categories with semidirect products, J. Pure Appl. Algebra 216 (2012), 1968-1975 · Zbl 1264.18004 |
| [58] | N. Martins-Ferreira, A. Montoli, M. Sobral: Semidirect products and crossed mod-ules in monoids with operations, J. Pure Appl. Algebra 217 (2013), 334-347 · Zbl 1277.18016 |
| [59] | N. Martins-Ferreira, A. Montoli, M. Sobral: Semidirect products and Split Short Five Lemma in normal categories, Applied Categ. Struct 22 (2014), 687-697 · Zbl 1311.18005 |
| [60] | J. C. C. McKinsey, A. Tarski: On closed elements in closure algebras, Ann. of Math. 47 (1946), 122-162 · Zbl 0060.06207 |
| [61] | B. Mitchell: Theory of Categories, Academic Press, New York and London, 1965 · Zbl 0136.00604 |
| [62] | D. Mundici: Interpretation of AFC ¦ algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63 · Zbl 0597.46059 |
| [63] | M. C. Pedicchio, E. M. Vitale: On the abstract characterization of quasi-varieties, Algebra univ. 43 (2000), 269-278 · Zbl 1011.08010 |
| [64] | D. Quillen: Higher algebraic K-theory, I, in: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Lecture Notes in Math. 341, Springer, Berlin, 1973 · Zbl 0292.18004 |
| [65] | L. Rédei: Die Verallgemeinerung der Schreierschen Erweiterungstheorie, Acta Sci. Math. (Szeged) 14 (1952), 252-273 · Zbl 0047.26602 |
| [66] | Z. Riečanová: Generalization of blocks for D-lattices and lattice-ordered effect al-gebras, Internat. J. Theoret. Phys. 39 (2000), no. 2, 231-237 · Zbl 0968.81003 |
| [67] | W. Rump: A decomposition theorem for square-free unitary solutions of the quan-tum Yang-Baxter equation, Adv. Math. 193 (2005), 40-55 · Zbl 1074.81036 |
| [68] | W. Rump: L-Algebras, Self-similarity, and l-Groups, J. Algebra 320 (2008), no. 6, 2328-2348 · Zbl 1158.06009 |
| [69] | W. Rump: A general Glivenko theorem, Algebra Univ. 61 (2009), no. 3-4, 455-473 · Zbl 1185.03097 |
| [70] | W. Rump: Right l-Groups, Geometric Garside Groups, and Solutions of the Quan-tum Yang-Baxter Equation, J. Algebra 439 (2015), 470-510 · Zbl 1360.20030 |
| [71] | W. Rump: Von Neumann algebras, L-algebras, Baer ¦ -monoids, and Garside groups, Forum Math. 30 (2018), no. 4, 973-995 · Zbl 1443.06005 |
| [72] | W. Rump: Commutative L-algebras and measure theory, Forum Math. 33, no. 6 (2021), 1527-1548 · Zbl 1500.28010 |
| [73] | W. Rump: Symmetric quantum sets and L-algebras, IMRN, Volume 2022, Issue 3, February 2022, Pages 1770-1810 doi:10.1093/imrn/rnaa135 · Zbl 1485.81039 · doi:10.1093/imrn/rnaa135 |
| [74] | W. Rump: L-algebras and three main non-classical logics, Ann. Pure Appl. Logic 173, Issue 7 (2022), 103121 · Zbl 1504.03040 |
| [75] | W. Rump: Structure groups of L-algebras and Hurwitz action, Geometriae Dedi-cata 216 (2022), DOI: 10.1007/s10711-022-00697-4 · Zbl 1497.57008 · doi:10.1007/s10711-022-00697-4 |
| [76] | J.-P. Schneiders: Quasi-abelian categories and sheaves, Mém. Soc. Math. France 1999, no. 76 · Zbl 0926.18004 |
| [77] | R. Succi Cruciani: La teoria delle relazioni nello studio di categorie regolari e di categorie esatte, Riv. Mat. Univ. Parma 1 (1975), 143-158 · Zbl 0356.18004 |
| [78] | A. Tarski: Grundzüge des Systemenkalküls, Erster Teil. In: Fundam. Math. 25 (1935), 503-526 · Zbl 0012.38501 |
| [79] | A. Ursini: Sulle varietà di algebre con buona teoria degli ideali, Boll. Unione Mat. Ital. 7 (1972), 90-95 · Zbl 0263.08006 |
| [80] | A. Ursini: On subtractive varieties, I, Algebra Univ. 31 (1994), 204-222 · Zbl 0799.08010 |
| [81] | E. M. Vitale: On the characterization of monadic categories over set, Cah. topol. géom. différentielle catég. 35, no. 4 (1994), 351-358 · Zbl 0824.18004 |
| [82] | Y. Wu, J. Wang, Y. Yang: Lattice-ordered effect algebras and L-algebras, Fuzzy Sets and Systems 369 (2019), 103-113 · Zbl 1423.03255 |
| [83] | Institute for Algebra and Number Theory, University of Stuttgart Pfaffenwaldring 57, D-70550 Stuttgart, Germany Email: rump@mathematik.uni-stuttgart.de |
| [84] | Maria Manuel Clementino, Universidade de Coimbra: mmc@mat.uc.pt Valeria de Paiva, Nuance Communications Inc: valeria.depaiva@gmail.com Richard Garner, Macquarie University: richard.garner@mq.edu.au Ezra Getzler, Northwestern University: getzler (at) northwestern(dot)edu |
| [85] | Dirk Hofmann, Universidade de Aveiro: dirk@ua.pt Joachim Kock, Universitat Autònoma de Barcelona: kock (at) mat.uab.cat Stephen Lack, Macquarie University: steve.lack@mq.edu.au Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Matias Menni, Conicet and Universidad Nacional de La Plata, Argentina: matias.menni@gmail.com Susan Niefield, Union College: niefiels@union.edu |
| [86] | Kate Ponto, University of Kentucky: kate.ponto (at) uky.edu Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosický, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Michael Shulman, University of San Diego: shulman@sandiego.edu Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si James Stasheff, University of North Carolina: jds@math.upenn.edu |
| [87] | Tim Van der Linden, Université catholique de Louvain: tim.vanderlinden@uclouvain.be |
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