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Abstrakty
The primary goal of the paper is to establish a duality for quasilattices. The main ingredients are duality for semilattices and their representations, the structural analysis of quasilattices as Płonka sums of lattices, and the duality for lattices developed by Hartonas and Dunn. Lattice duality treats the identity function on a lattice as a Galois connection between its meet and join semilattice reducts, and then invokes a duality between Galois connections and polarities. A second goal of the paper is a further examination of this latter duality, using the concept of a pairing to provide an algebraic equivalent to the relational structure of a polarity.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
Strony
331--359
Opis fizyczny
Bibliogr. 33 poz., rys.
Twórcy
autor
- Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warszawa, Poland
autor
- Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA
Bibliografia
- [1] Płonka J. On distributive quasilattices. Fundamenta Mathematicae, 1967;60(2):191–200.
- [2] Padmanabhan R. Regular identities in lattices. Transactions of the American Mathematical Society, 1971;158(1):179–188. URL http://www.jstor.org/stable/1995780.
- [3] Harding J, Romanowska A. Varieties of Birkhoff systems I. Order, 2017;34(1):45–68, URL https://doi:10.1007/s11083-016-9388-x.
- [4] Harding J, Romanowska A. Varieties of Birkhoff systems II. Order, 2017;34(1):69–89, URL https://doi:10.1007/s11083-016-9392-1.
- [5] Gierz G, Romanowska A. Duality for distributive bisemilattices. J. Australian Math. Soc., 1991;51:247–275. URL https://doi.org/10.1017/S1446788700034224.
- [6] Hartonas C, Dunn JM, Stone duality for lattices. Algebra Universalis, 1997;37:391–401. URL https://doi.org/10.1007/s000120050024.
- [7] Urquhart A. A topological representation theory for lattices. Algebra Universalis, 1978;8:45–58.
- [8] Hartung G. A topological representation of lattices. Algebra Universalis, 1992;29(2):273–299. URL https://doi.org/10.1007/BF01190610.
- [9] Wille, R. Restructuring lattice theory: an approach based on hierarchies of concepts. In Ordered sets (Banff, AB, 1981). NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. Dordrecht: Reidel, 1982;83:445-470. URL https://doi.org/10.1007/978-94-009-7798-3_15.
- [10] Hofmann KH, Mislove M, Stralka A. The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications. Berlin: Springer Lecture Notes in Mathematics, vol. 396; 1974. ISBN: 978-354006807-5.
- [11] Romanowska A, Smith J. Semilattice-based dualities. Studia Logica, 1996;56(1–2):225–261. URL https://doi.org/10.1007/BF00370148.
- [12] Romanowska A, Smith J. Duality for semilattice representations. Journal of Pure and Applied Algebra, 1997;115(3):289–308. URL https://doi.org/10.1016/S0022-4049(96)00026-6.
- [13] Romanowska A, Smith J. Post-Modern Algebra. New York (NY): Wiley, 1999.
- [14] Caramello, O. A topos-theoretic approach to Stone-type dualities, 2011. arXiv:1103.3493v1 [math.CT].
- [15] Davey, BA, Werner H. Dualities and equivalences for varieties of algebras. Colloq. Math. Soc. J. Bolyai, 1980;33:101–275.
- [16] Davey BA. Duality theory on ten dollars a day. 1992. La Trobe University Mathematics Research Paper No. 131 92-3, Melbourne.
- [17] Hartonas C, Dunn, JM Duality theorems for partial orders, semilattices, Galois connections and lattices. 1993. Indiana University Logic Group Preprint No. IULG 93-26. Bloomington (IN). URL https://www.researchgate.net/publication/2573090.
- [18] Rasiowa H, Sikorski R. The Mathematics of Metamathematics (3rd. edition). Warsaw, PWN, 1970.
- [19] Rasiowa H. An Algebraic Approach to Non-Classical Logics. Amsterdam: North-Holland, 1974.
- [20] Orłowska E, Rewitzky I. Algebras for Galois-style connections and their discrete duality. Fuzzy Sets and Systems, 2010;161(9):1325–1342. URL https://doi.org/10.1016/j.fss.2009.12.013.
- [21] Hardegree GM. (1982). An approach to the logic of natural kinds. Pacific Phil. Quarterly, 1982;63:122–132. URL doi:10.1111/j.1468-0114.1982.tb00093.x.
- [22] Birkhoff G. Lattice Theory (3rd. edition). Providence (RI): American Mathematical Society, 1967. ISBN:978-0-8218-1025-5. URL http://bookstore.ams.org/coll-25.
- [23] Herrlich H, Hušek M. Galois connections. In Melton A, editor, Mathematical Foundations of Programming Semantics (Manhattan, KS, 1985). Springer Lecture Notes in Computer Science, vol. 239. Springer, Berlin, 1986 pp 122–134. ISBN: 3-540-16816-8. URL https://link.springer.com/chapter/10.1007/3-540-16816-8_28.
- [24] Herrlich H, Strecker GE Category Theory. Boston (MA): Allyn and Bacon, 1973. ISBN: 978-3-88538-001-6. URL http://www.heldermann.de/SSPM/SSPM01/sspm01.htm.
- [25] Mac Lane S. Categories for the Working Mathematician. Berlin: Springer, 1971. ISBN: 978-1-4757-4721-8. URL http://www.springer.com/us/book/9780387984032.
- [26] Johnstone PT. Stone Spaces. Cambridge: Cambridge University Press, 1982. ISBN: 978-0-521-33779-3.
- [27] Mac Lane S, Moerdijk I. Sheaves in Geometry and Logic. New York (NY): Springer, 1992. ISBN: 978-0-387-97710-2. URL http://www.springer.com/us/book/9780387977102.
- [28] Caramello O. Theories, Sites, Toposes. Oxford: Oxford University Press, 2018. ISBN: 978-0-198-75891-4. URL https://global.oup.com/academic/product/theories-sites-toposes-9780198758914.
- [29] Howie JM. An Introduction to Semigroup Theory. London: Academic Press, 1976. ISBN: 978-0-127-54633-9.
- [30] Romanowska A, Smith J. Modal Theory. Berlin: Heldermann, 1985. ISBN: 3-88538-209-1. URL http://www.heldermann.de/R&E/raecont.htm#vol9.
- [31] Płonka J. On a method of construction of abstract algebras. Fundamenta Mathematicae, 1967;61(2):183–189. URL http://eudml.org/doc/213988.
- [32] Płonka J, Romanowska A. Semilattice sums. In Romanowska A, Smith, J, editors, Universal Algebra and Quasigroup Theory, Heldermann, Berlin, 1992 pp. 123–158. ISBN: 3-88538-219-9. URL http://www.heldermann.de/R&E/raecont.htm#vol19.
- [33] Romanowska A, Smith J. Modes. River Edge (NJ): World Scientific, 2002. ISBN: 978-981-02-4942-7. URL http://www.worldscientific.com/worldscibooks/10.1142/4953.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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