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Type Monoids and V-Measures

  • Friedrich Wehrung
Chapter
  • 458 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2188)

Abstract

The type monoid of a Boolean inverse semigroup is an abstraction of the concept of monoid of equidecomposability types of a Boolean ring under a group action. The latter concept has been studied in a wide array of works including Banach [17], Tarski [109]. Its relation with type monoids of Boolean inverse semigroups was recognized in Wallis’ Ph.D. thesis [116], see also Kudryavtseva et al. [71], Lawson and Scott [77].

Bibliography

  1. 8.
    Ara, P., Facchini, A.: Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math. 18(3), 365–389 (2006). MR 2237927 (2007d:16012)Google Scholar
  2. 17.
    Banach, S.: Un théorème sur les transformations biunivoques. Fund. Math. 6(1), 236–239 (1924) (French)Google Scholar
  3. 19.
    Barwise, J.: Admissible Sets and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic. Springer, Berlin/New York (1975). MR 0424560 (54 #12519)Google Scholar
  4. 32.
    Dobbertin, H.: On Vaught’s criterion for isomorphisms of countable Boolean algebras. Algebra Univers. 15(1), 95–114 (1982). MR 663956 (83m:06017)Google Scholar
  5. 33.
    Dobbertin, H.: Refinement monoids, Vaught monoids, and Boolean algebras. Math. Ann. 265(4), 473–487 (1983). MR 721882 (85e:06016)Google Scholar
  6. 34.
    Dobbertin, H.: Measurable refinement monoids and applications to distributive semilattices, Heyting algebras, and Stone spaces. Math. Z. 187(1), 13–21 (1984). MR 753415 (85h:20072)Google Scholar
  7. 35.
    Dobbertin, H.: Vaught measures and their applications in lattice theory. J. Pure Appl. Algebra 43(1), 27–51 (1986). MR 862871 (87k:06032)Google Scholar
  8. 58.
    Hanf, W.: Primitive Boolean algebras. In: Proceedings of the Tarski Symposium (Proceedings of the Symposia Pure Mathematics, University of California, Berkeley, CA, 1971), vol. 25, pp. 75–90. American Mathematical Society, Providence, RI (1974). MR 0379182 (52 #88)Google Scholar
  9. 60.
    Howie, J.M.: An Introduction to Semigroup Theory. London Mathematical Society Monographs, vol. 7. Academic/Harcourt Brace Jovanovich Publishers, London/New York (1976). MR 0466355 (57 #6235)Google Scholar
  10. 67.
    Karp, C.R.: Finite-quantifier equivalence. In: Theory of Models (Proceedings of the 1963 International Symposium at Berkeley), pp. 407–412. North-Holland, Amsterdam (1965). MR 0209132 (35 #36)Google Scholar
  11. 71.
    Kudryavtseva, G., Lawson, M.V., Lenz, D.H., Resende, P.: Invariant means on Boolean inverse monoids. Semigroup Forum 92(1), 77–101 (2016). MR 3448402Google Scholar
  12. 72.
    Lawson, M.V.: Enlargements of regular semigroups. Proc. Edinb. Math. Soc. (2) 39(3), 425–460 (1996). MR 1417688 (97k:20104)Google Scholar
  13. 76.
    Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. Adv. Math. 244, 117–170 (2013). MR 3077869Google Scholar
  14. 77.
    Lawson, M.V., Scott, P.: AF inverse monoids and the structure of countable MV-algebras. J. Pure Appl. Algebra 221(1), 45–74 (2017). MR 3531463Google Scholar
  15. 88.
    Nambooripad, K.S.S.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. (2) 23(3), 249–260 (1980). MR 620922 (82g:20092)Google Scholar
  16. 94.
    Pierce, R.S.: Countable Boolean algebras. In: Handbook of Boolean Algebras, vol. 3, pp. 775–876. North-Holland, Amsterdam (1989). MR 991610Google Scholar
  17. 109.
    Tarski, A.: Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni Jónsson and Alfred Tarski. Oxford University Press, New York, NY (1949). MR 0029954 (10,686f)Google Scholar
  18. 114.
    Vaught, R.L.: Topics in the theory of arithmetical classes and Boolean algebras. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.). University of California, Berkeley (1955). MR 2938637Google Scholar
  19. 116.
    Wallis, A.R.: Semigroup and category-theoretic approaches to partial symmetry, Ph.D. thesis. Heriot-Watt University, Edinburgh (2013)Google Scholar
  20. 118.
    Wehrung, F.: Injective positively ordered monoids. I. J. Pure Appl. Algebra 83(1), 43–82 (1992). MR 1190444 (93k:06023)Google Scholar
  21. 122.
    Wehrung, F.: The dimension monoid of a lattice. Algebra Univers. 40(3), 247–411 (1998). MR 1668068 (2000i:06014)Google Scholar
  22. 124.
    Wehrung, F.: Non-measurability properties of interpolation vector spaces. Israel J. Math. 103, 177–206 (1998). MR 1613568 (99g:06023)Google Scholar
  23. 125.
    Wehrung, F.: A K 0-avoiding dimension group with an order-unit of index two. J. Algebra 301(2), 728–747 (2006). MR 2236765 (2007e:16011)Google Scholar
  24. 126.
    Zhitomirskiy, G.I.: Inverse semigroups and fiberings. In: Semigroups (Luino, 1992), pp. 311–321. World Scientific Publishing, River Edge, NJ (1993). MR 1647275Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  • Friedrich Wehrung
    • 1
  1. 1.Département de MathématiquesUniversité de Caen NormandieCaenFrance

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