Some remarks on distributive semilattices

Sergio A. Celani; Ismael Calomino

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 3, page 407-428
  • ISSN: 0010-2628

Abstract

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In this paper we shall give a survey of the most important characterizations of the notion of distributivity in semilattices with greatest element and we will present some new ones through annihilators and relative maximal filters. We shall also simplify the topological representation for distributive semilattices given in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, and show that the meet-relations are closed under composition. So, we obtain that the D S -spaces with meet-relations is a category dual to the category of distributive semilattices with homomorphisms. These results complete the topological representation presented in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, without the use of ordered topological spaces. Finally, following the work of G. Bezhanishvili and R. Jansana in Generalized Priestley quasi-orders, Order 28 (2011), 201–220, we will prove a characterization of homomorphic images of a distributive semilattice A by means of family of closed subsets of the dual space endowed with a lower Vietoris topology.

How to cite

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Celani, Sergio A., and Calomino, Ismael. "Some remarks on distributive semilattices." Commentationes Mathematicae Universitatis Carolinae 54.3 (2013): 407-428. <http://eudml.org/doc/260644>.

@article{Celani2013,
abstract = {In this paper we shall give a survey of the most important characterizations of the notion of distributivity in semilattices with greatest element and we will present some new ones through annihilators and relative maximal filters. We shall also simplify the topological representation for distributive semilattices given in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, and show that the meet-relations are closed under composition. So, we obtain that the $DS$-spaces with meet-relations is a category dual to the category of distributive semilattices with homomorphisms. These results complete the topological representation presented in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, without the use of ordered topological spaces. Finally, following the work of G. Bezhanishvili and R. Jansana in Generalized Priestley quasi-orders, Order 28 (2011), 201–220, we will prove a characterization of homomorphic images of a distributive semilattice $A$ by means of family of closed subsets of the dual space endowed with a lower Vietoris topology.},
author = {Celani, Sergio A., Calomino, Ismael},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distributive semilattices; topological representation; meet-relations; distributive semilattices; topological representation; meet-relations},
language = {eng},
number = {3},
pages = {407-428},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some remarks on distributive semilattices},
url = {http://eudml.org/doc/260644},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Celani, Sergio A.
AU - Calomino, Ismael
TI - Some remarks on distributive semilattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 3
SP - 407
EP - 428
AB - In this paper we shall give a survey of the most important characterizations of the notion of distributivity in semilattices with greatest element and we will present some new ones through annihilators and relative maximal filters. We shall also simplify the topological representation for distributive semilattices given in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, and show that the meet-relations are closed under composition. So, we obtain that the $DS$-spaces with meet-relations is a category dual to the category of distributive semilattices with homomorphisms. These results complete the topological representation presented in Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51, without the use of ordered topological spaces. Finally, following the work of G. Bezhanishvili and R. Jansana in Generalized Priestley quasi-orders, Order 28 (2011), 201–220, we will prove a characterization of homomorphic images of a distributive semilattice $A$ by means of family of closed subsets of the dual space endowed with a lower Vietoris topology.
LA - eng
KW - distributive semilattices; topological representation; meet-relations; distributive semilattices; topological representation; meet-relations
UR - http://eudml.org/doc/260644
ER -

References

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  1. Balbes R., 10.1090/S0002-9947-1969-0233741-7, Trans. Amer. Math. Soc. 136 (1969), 261–267. Zbl0175.01402MR0233741DOI10.1090/S0002-9947-1969-0233741-7
  2. Bezhanishvili G., Jansana R., 10.1007/s11225-011-9323-5, Studia Logica 98 (2011), 83–122. Zbl1238.03050MR2821271DOI10.1007/s11225-011-9323-5
  3. Bezhanishvili G., Jansana R., 10.1007/s11083-010-9166-0, Order 28 (2011), 201–220. Zbl1243.06003MR2819220DOI10.1007/s11083-010-9166-0
  4. Celani S.A., Topological representation of distributive semilattices, Sci. Math. Japonicae online 8 (2003), 41–51. Zbl1041.06002MR1987817
  5. Chajda I., Halaš R., Kühr J., Semilattice Structures, Research and Exposition in Mathematics, 30, Heldermann Verlag, Lemgo, 2007. MR2326262
  6. Cornish W.H., Hickman R.C., 10.1007/BF01902195, Acta Math. Acad. Sci. Hungar. 32 (1978), 5–16. Zbl0497.06005MR0551490DOI10.1007/BF01902195
  7. Grätzer G., General Lattice Theory, Birkhäuser, Basel, 1998. MR1670580
  8. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M.W., Scott D.S., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003. Zbl1088.06001MR1975381
  9. Hickman R.C., 10.1017/S1446788700025374, J. Austral. Math. Soc. Ser. A 36 (1984), 287–315. Zbl0537.06006MR0733904DOI10.1017/S1446788700025374
  10. Mandelker M., 10.1215/S0012-7094-70-03748-8, Duke Math. J. 37 (1970), 377–386. Zbl0206.29701MR0256951DOI10.1215/S0012-7094-70-03748-8
  11. Priestley H.A., 10.1112/blms/2.2.186, Bull. London Math. Soc. 2 (1970), 186–190. Zbl0201.01802MR0265242DOI10.1112/blms/2.2.186
  12. Rhodes J.B., 10.1090/S0002-9947-1975-0351935-X, Trans. Amer. Math. Soc. 201 (1975), 31–41. Zbl0326.06006MR0351935DOI10.1090/S0002-9947-1975-0351935-X
  13. Stone M., Topological representation of distributive lattices and Brouwerian logics, Časopis pěst. mat. fys. 67 (1937), 1–25. 
  14. Varlet J.C., A generalization of the notion of pseudo-complementedness, Bull. Soc. Roy. Sci. Liège 37 (1968), 149–158. Zbl0162.03501MR0228390
  15. Varlet J.C., Distributive semilattices and Boolean lattices, Bull. Soc. Roy. Liège 41 (1972), 5–10. Zbl0302.06025MR0307991
  16. Varlet J.C., 10.1017/S0004972700043094, Bull. Austral. Math. Soc. 9 (1973), 169–185. Zbl0258.06009MR0382091DOI10.1017/S0004972700043094
  17. Varlet J.C., 10.1007/BF02194889, Semigroup Forum 10 (1975), 220–228. Zbl0299.06002MR0384627DOI10.1007/BF02194889

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