<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 5 Issue 3" %>
A Combinatorial Representation for a Special Class of Complete Distributive Lattices
Luca Ferrari1and Giorgio Nicoletti2
1Dipartimento di Matematica "U. Dini", Viale Morgagni 67/A, 50135 Firenze, Italy
ferrari@math.unifi.it
2Dipartimento di Matematica per le Scienze Economiche e Sociali, Piazza Scaravilli 2, 40126 Bologna, Italy
preside@ipazia.economia.unibo.it
Annals of Combinatorics 5 (3) p.285-304 September, 2001
AMS Subject Classification: 06D05, 06D50
Abstract:
We prove a representation theorem for a special class of bounded distributive lattices by making use of purely combinatorial and lattice-theoretical techniques only. Some particular cases are discussed.
Keywords: infinitely distributive lattices, duality, P-lattices, completely irreducible elements

References:

1.  H. Crapo, Unities and negation: on the representation of finite lattices, J. Pure Appl. Algebra 23 (1982) 109–135.

2.  H. Crapo and C. Le Conte de Poly-Barbut, Unities and negation, In: Mathematical Essays in Honor of Gian-Carlo Rota, Birkhäuser, Boston, 1998, pp. 131–155.

3.  P. Crawley and R.P. Dilworth, Algebraic Theory of Lattices, Prentice-Hall, N.J., 1973.

4.  B.A. Davey, On the lattice of subvarieties, Houston J. Math. 5 (1979) 183–192.

5.  B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990.

6.  M. Funayama and T. Nakayama, On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo 18 (1942) 553–554.,

7.  G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1978.

8.  K.H. Hofmann, M. Mislove, and A. Stralka, The Pontryagin Duality of Compact 0- Dimensional Semilattices and Its Applications, Lecture Notes in Mathematics, Vol. 396, Springer-Verlag, Berlin, Heidelberg, New York, 1974.

9.  W.A. Lampe, A perspective on algebraic representations of lattices, Algebra Univ. 31 (1994) 337–364.

10.  L. Pezzoli, On D-complementation, Adv. Math. 51 (1984) 226–239.

11.  G.N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953) 518–522.

12.  V.N. Saliï , Lattices with Unique Complements, Translations of Mathematical Monographs, Vol. 69, Amer. Math. Soc., Providence, Rhode Island, 1988.

13.  G. Szasz, Introduction to Lattice Theory, Academic Press, New York and London, 1963.