On coarse anti-Lawson semilattices

V. L. Frider; M. M. Zarichnyi

The category of the anti-Lawson semilattices is isomorphic to that of the algebras for the finite hyperspace monad in the coarse category. A notion of anti-Lawson coarse semilattice is introduced. We construct an example of a coarse semilattice which is not an anti-Lawson coarse semilattice. It is proved also that every coarse semilattice of asymptotic dimension (in the sense of Gromov) zero is an anti-Lawson coarse semilattice.

1. Wyler O. Algebraic theories of continuous semilattices, Arch. Ration. Mech. Anal. 90 (1985), 99–113.

2. Harvey J.M. Categorical caracterization of uniform hyperspaces, Math. Proc. Camb. Phil. Soc. (1983), no. 94, 229–233.

3. Higson N.; Pedersen E. K.; Roe J. $C^*$-algebras and controlled topology. $K$-Theory 11 (1997), no. 3, 209--239.

4. Frider V., Zarichnyi M. Hyperspace functor in the coarse category, Visn. L'viv. un-tu. Ser. Mech.-Mat. (2003), № 61, 78–86.

5. Lawson L. Embeddings of compact convex sets and locally compact cones, Pacif. J. Math. 66 (1976), 443–453.

6. Gierz G. A compact semilattice on the Hilbert cube with no interval homomorphism, Proc. Amer. Math. Soc. 104 (1987), 592–594.

7. Lawson L. Lattices with no interval homomorphisms, Pacif. J. Math. 32 (1970), 459–465.

8. Mitchener P. D. Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271–297.

9. Roe J. Index Theory, Coarse Geometry, and Topology of Manifolds, CBMS regional Conference Series in Mathematics, no. 90, 1996.

10. Skandalis S., Tu J. L., Yu G. Coarse Baum-Connes conjecture and groupoids, Preprint.

11. Higson N., Roe J. Analytic $K$-Homology, Oxford University Press, 2000.

12. Roberts J.W. A compact convex set with no extreme points, Studia Math. 60 (1977), 255–266.

13. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D. A compendium of continuous lattices. Berlin e.a.: Springer, 1980.

14. Gromov M. Asymptotic invariants for infinite groups, LMS Lecture Notes, 182 (2), 1993.

15. Protasov I.V. Metrizable ball structures, Algebra and Discrete Math. 1 (2002), 1–13.

16. Dranishnikov A., Asymptotic topology. Uspekhi Mat. Nauk 55 (2000), no. 6, 71–116 (Russian); translation in Russian Math. Surveys 55 (2000), no. 6, 1085–1129.

	On coarse anti-Lawson semilattices
Author	V. L. Frider, M. M. Zarichnyi topology@franko.lviv.ua Lviv Ivan Franko National University,Faculty of Mechanics and Mathematics
Abstract	The category of the anti-Lawson semilattices is isomorphic to that of the algebras for the finite hyperspace monad in the coarse category. A notion of anti-Lawson coarse semilattice is introduced. We construct an example of a coarse semilattice which is not an anti-Lawson coarse semilattice. It is proved also that every coarse semilattice of asymptotic dimension (in the sense of Gromov) zero is an anti-Lawson coarse semilattice.
Keywords	anti-Lawson semilattice, finite hyperspace monad, coarse category
DOI	doi:10.30970/ms.21.1.3-12
Reference	1. Wyler O. Algebraic theories of continuous semilattices, Arch. Ration. Mech. Anal. 90 (1985), 99–113. 2. Harvey J.M. Categorical caracterization of uniform hyperspaces, Math. Proc. Camb. Phil. Soc. (1983), no. 94, 229–233. 3. Higson N.; Pedersen E. K.; Roe J. $C^*$-algebras and controlled topology. $K$-Theory 11 (1997), no. 3, 209--239. 4. Frider V., Zarichnyi M. Hyperspace functor in the coarse category, Visn. L'viv. un-tu. Ser. Mech.-Mat. (2003), № 61, 78–86. 5. Lawson L. Embeddings of compact convex sets and locally compact cones, Pacif. J. Math. 66 (1976), 443–453. 6. Gierz G. A compact semilattice on the Hilbert cube with no interval homomorphism, Proc. Amer. Math. Soc. 104 (1987), 592–594. 7. Lawson L. Lattices with no interval homomorphisms, Pacif. J. Math. 32 (1970), 459–465. 8. Mitchener P. D. Coarse homology theories, Algebr. Geom. Topol. 1 (2001), 271–297. 9. Roe J. Index Theory, Coarse Geometry, and Topology of Manifolds, CBMS regional Conference Series in Mathematics, no. 90, 1996. 10. Skandalis S., Tu J. L., Yu G. Coarse Baum-Connes conjecture and groupoids, Preprint. 11. Higson N., Roe J. Analytic $K$-Homology, Oxford University Press, 2000. 12. Roberts J.W. A compact convex set with no extreme points, Studia Math. 60 (1977), 255–266. 13. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D. A compendium of continuous lattices. Berlin e.a.: Springer, 1980. 14. Gromov M. Asymptotic invariants for infinite groups, LMS Lecture Notes, 182 (2), 1993. 15. Protasov I.V. Metrizable ball structures, Algebra and Discrete Math. 1 (2002), 1–13. 16. Dranishnikov A., Asymptotic topology. Uspekhi Mat. Nauk 55 (2000), no. 6, 71–116 (Russian); translation in Russian Math. Surveys 55 (2000), no. 6, 1085–1129.
Pages	3-12
Volume	21
Issue	1
Year	2004
Journal	Matematychni Studii
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