The prime spectrum of an $L$-algebra
HTML articles powered by AMS MathViewer
- by Wolfgang Rump and Leandro Vendramin;
- Proc. Amer. Math. Soc. 152 (2024), 3197-3207
- DOI: https://doi.org/10.1090/proc/16802
- Published electronically: June 5, 2024
- HTML | PDF | Request permission
Abstract:
We prove that the lattice of ideals of an arbitrary $L$-algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ideals) of $L$-algebras and characterize prime ideals in topological terms.References
- E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101â126. MR 19087, DOI 10.2307/1969218
- Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 552653, DOI 10.1007/BFb0067004
- Garrett Birkhoff and John von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823â843. MR 1503312, DOI 10.2307/1968621
- W. J. Blok and I. M. A. Ferreirim, Hoops and their implicational reducts (abstract), Algebraic methods in logic and in computer science (Warsaw, 1991) Banach Center Publ., vol. 28, Polish Acad. Sci. Inst. Math., Warsaw, 1993, pp. 219â230. MR 1446285
- Egbert Brieskorn and Kyoji Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245â271 (German). MR 323910, DOI 10.1007/BF01406235
- C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467â490. MR 94302, DOI 10.1090/S0002-9947-1958-0094302-9
- R. Cignoli, D. Gluschankof, and F. Lucas, Prime spectra of lattice-ordered abelian groups, J. Pure Appl. Algebra 136 (1999), no. 3, 217â229. MR 1675803, DOI 10.1016/S0022-4049(98)00031-0
- Michael R. Darnel, Theory of lattice-ordered groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 187, Marcel Dekker, Inc., New York, 1995. MR 1304052
- Patrick Dehornoy, Groupes de Garside, Ann. Sci. Ăcole Norm. Sup. (4) 35 (2002), no. 2, 267â306 (French, with English and French summaries). MR 1914933, DOI 10.1016/S0012-9593(02)01090-X
- Patrick Dehornoy, François Digne, Eddy Godelle, Daan Krammer, and Jean Michel, Foundations of Garside theory, EMS Tracts in Mathematics, vol. 22, European Mathematical Society (EMS), ZĂŒrich, 2015. Author name on title page: Daan Kramer. MR 3362691, DOI 10.4171/139
- Patrick Dehornoy and Luis Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3) 79 (1999), no. 3, 569â604. MR 1710165, DOI 10.1112/S0024611599012071
- Pierre Deligne, Les immeubles des groupes de tresses gĂ©nĂ©ralisĂ©s, Invent. Math. 17 (1972), 273â302 (French). MR 422673, DOI 10.1007/BF01406236
- Anatolij DvureÄenskij and Sylvia PulmannovĂĄ, New trends in quantum structures, Mathematics and its Applications, vol. 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000. MR 1861369, DOI 10.1007/978-94-017-2422-7
- Pavel Etingof, Travis Schedler, and Alexandre Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), no. 2, 169â209. MR 1722951, DOI 10.1215/S0012-7094-99-10007-X
- F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235â254. MR 248801, DOI 10.1093/qmath/20.1.235
- Roberto Giuntini and Heinz Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 (1989), no. 7, 931â945. MR 1013913, DOI 10.1007/BF01889307
- George GrÀtzer, General lattice theory, BirkhÀuser Verlag, Basel, 2003. With appendices by B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung and R. Wille; Reprint of the 1998 second edition [MR1670580]. MR 2451139
- Stanley Gudder, Sharp and unsharp quantum effects, Adv. in Appl. Math. 20 (1998), no. 2, 169â187. MR 1601375, DOI 10.1006/aama.1997.0575
- M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43â60. MR 251026, DOI 10.1090/S0002-9947-1969-0251026-X
- Zurab Janelidze, The pointed subobject functor, $3\times 3$ lemmas, and subtractivity of spans, Theory Appl. Categ. 23 (2010), No. 11, 221â242. MR 2720189
- G. Janelidze, L. Marki, W. Tholen, and A. Ursini, Ideal determined categories, Cah. Topol. GĂ©om. DiffĂ©r. CatĂ©g. 51 (2010), no. 2, 115â125 (English, with English and French summaries). MR 2667979
- Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
- Peter Köhler, Brouwerian semilattices, Trans. Amer. Math. Soc. 268 (1981), no. 1, 103â126. MR 628448, DOI 10.1090/S0002-9947-1981-0628448-3
- Tomasz Kowalski, A syntactic proof of a conjecture of Andrzej WroĆski, Rep. Math. Logic 28 (1994), 81â86 (1995). MR 1411401
- W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1971. MR 511676
- Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic, Universitext, Springer-Verlag, New York, 1994. A first introduction to topos theory; Corrected reprint of the 1992 edition. MR 1300636, DOI 10.1007/978-1-4612-0927-0
- J. C. C. McKinsey and Alfred Tarski, On closed elements in closure algebras, Ann. of Math. (2) 47 (1946), 122â162. MR 15037, DOI 10.2307/1969038
- Daniele Mundici, Interpretation of AF $C^\ast$-algebras in Ćukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15â63. MR 819173, DOI 10.1016/0022-1236(86)90015-7
- Zdenka RieÄanovĂĄ, Generalization of blocks for $D$-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39 (2000), no. 2, 231â237. MR 1762594, DOI 10.1023/A:1003619806024
- Wolfgang Rump, $L$-algebras, self-similarity, and $l$-groups, J. Algebra 320 (2008), no. 6, 2328â2348. MR 2437503, DOI 10.1016/j.jalgebra.2008.05.033
- Wolfgang Rump, A general Glivenko theorem, Algebra Universalis 61 (2009), no. 3-4, 455â473. MR 2565867, DOI 10.1007/s00012-009-0018-y
- Wolfgang Rump, Right $l$-groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation, J. Algebra 439 (2015), 470â510. MR 3373381, DOI 10.1016/j.jalgebra.2015.04.045
- Wolfgang Rump, Von Neumann algebras, $L$-algebras, Baer $^*$-monoids, and Garside groups, Forum Math. 30 (2018), no. 4, 973â995. MR 3824801, DOI 10.1515/forum-2017-0108
- Wolfgang Rump, $L$-algebras with duality and the structure group of a set-theoretic solution to the Yang-Baxter equation, J. Pure Appl. Algebra 224 (2020), no. 8, 106314, 12. MR 4074571, DOI 10.1016/j.jpaa.2020.106314
- W. Rump, Symmetric quantum sets, Int. Math. Res. Not. IMRN (2022), no. 3, 1770â1810.
- Wolfgang Rump, $L$-algebras and three main non-classical logics, Ann. Pure Appl. Logic 173 (2022), no. 7, Paper No. 103121, 25. MR 4405373, DOI 10.1016/j.apal.2022.103121
- Wolfgang Rump, $L$-algebras and topology, J. Algebra Appl. 22 (2023), no. 2, Paper No. 2350034, 23. MR 4541621, DOI 10.1142/S0219498823500342
- M. H. Stone, Topological representation of distributive lattices and Brouwerian logics, Äasopis PeĆĄt. Mat. Fys. 67 (1937), 1â25
- Andrzej WroĆski, An algebraic motivation for BCK-algebras, Math. Japon. 30 (1985), no. 2, 187â193. MR 795873
Bibliographic Information
- Wolfgang Rump
- Affiliation: Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
- MR Author ID: 226306
- Email: rump@mathematik.uni-stuttgart.de
- Leandro Vendramin
- Affiliation: Department of Mathematics and Data Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel
- MR Author ID: 829575
- ORCID: 0000-0003-0954-7785
- Email: Leandro.Vendramin@vub.be
- Received by editor(s): June 2, 2022
- Received by editor(s) in revised form: January 8, 2024
- Published electronically: June 5, 2024
- Additional Notes: The second author was supported in part by OZR3762 of Vrije Universiteit Brussel and FWO Senior Research Project G004124N
- Communicated by: Jerzy Weyman
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3197-3207
- MSC (2020): Primary 06B10, 06F05, 08A55
- DOI: https://doi.org/10.1090/proc/16802
- MathSciNet review: 4767255
Dedicated: to B. V. M.