Stone compactification of additive generalized-algebraic lattices


  • Xueyou Chen Shandong University of Technology
  • Quingguo Li Hunan University
  • Zike Deng Hunan University



Additivity, Generalized way-below relation, Lower homomorphism, Upper adjoint


In this paper, the notions of regular, completely regular, compact additive generalized algebraic lattices are introduced, and Stone compactification is constructed. The following theorem is also obtained.

Theorem: An additive generalized algebraic lattice has a Stone compactification if and only if it is regular and completely regular.


Download data is not yet available.

Author Biographies

Xueyou Chen, Shandong University of Technology

College of Mathematics and Information Science

Quingguo Li, Hunan University

College of Mathematics and Economics

Zike Deng, Hunan University

College of Mathematics and Economics


H. J. Bandelt, M-distributive lattices, Arch Math 39 (1982), 436–444.

X. Chen, Q. Li, F. Long and Z. Deng, Tietze Extension Theorem on Additive Generalized Algebraic Lattice, Acta. Mathematica Scientia (A)(in Chinese), accepted.

Z. Deng, Generalized-continuous lattices I, J. Hunan Univ. 23, No. 3 (1996), 1–3.

Z. Deng, Generalized-continuous lattices II, J. Hunan Univ. 23, No. 5 (1996), 1–3.

Z. Deng, Homomorphisms of generalized-continuous lattices, J. Hunan Univ. 26, No. 3 (1999), 1–4.

Z. Deng, Direct sums and sublattices of generalized-continuous lattices, J. Hunan Univ. 28, No. 1 (2001), 1–4.

Z. Deng, Topological representation for generalized-algebraic lattices, (in W.Charles. Holland, edited: Ordered Algebraic structures, Algebra, Logic and Applications Vol 16, 49-55 Gordon and breach Science publishers, 2001.)

Z. Deng, Additivity of generalized algebraic lattices and T0-topology, J. Hunan Univ. 29, No. 5 (2002), 1–3.

Z. Deng, Structures of generalized-continuous lattices, J. Hunan Univ. 28, 6 (2001), 1–4.

Z. Deng, Representation of strongly generalized-continuous lattices in terms of complete chains, J. Hunan Univ. 29, No. 3 (2002), 8–10.

D. Drake and W. J. Thron, On representation of an abstract lattice as the family of closed sets of a topological space, Trans. Amer. Math. Soc. 120 (1965), 57–71.

G. Gierz et, al, A Compendium of Continuous Lattices, Berlin, Speringer- Verlag, 1980.

P. T. Johnstone, Stone Spaces, Cambridge Univ press, Cambridge, 1983.

J. L. Kelly, General Topology, Van Nostrand Princeton, NJ, 1995.

D. Novak, Generalization of continuous posets, Trans. Amer. Math. Soc 272 (1982), 645–667.

S. Papert, Which distributive lattices are lattices of closed sets?, Proc. Cambridge. Phil. Soc. 55 (1959), 172–176.

Q. X. Xu, Construction of homomorphisms of M-continuous lattices, Trans. Amer. Math. Soc. 347 (1995), 3167–3175.


How to Cite

X. Chen, Q. Li, and Z. Deng, “Stone compactification of additive generalized-algebraic lattices”, Appl. Gen. Topol., vol. 8, no. 2, pp. 309–317, Oct. 2007.



Regular Articles