Document Type : Research Paper


1 Department of Mathematics, University of Hormozgan, Bandarabbas, Iran.

2 Department of Mathematics, Sirjan University of Technology, Sirjan, Iran.


In this paper, we introduce the concept of the generalized topological molecular lattices as a generalization of Wang's topological molecular lattices,  topological spaces, fuzzy topological spaces, L-fuzzy topological spaces and soft topological spaces. Topological molecular lattices were defined by closed elements, but in this new structure we present the concept of the open elements and define a closed element by the pseudocomplement of an open element.  We have two structures on a completely distributive complete lattice, topology and generalized co-topology which are not dual to each other. We study the basic concepts, in particular separation axioms and some relations among them.


Main Subjects

[1] A.R. Aliabadi and A. Sheykhmiri, LG-topology, Bull. Iranian Math. Soc., 41 (2015), pp. 239-258.
[2] T.S. Blyth, Lattice and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
[3] G. Bruns, Darstellungen und Erweiterungen geordneter Mengen II, J. Reine Angew. Math., 210 (1962), pp. 1-23.
[4] K. El-Saady and F. Al-Nabbat, Generalized topological molecular lattices, Advances in Pure Mathematics, 5 (2015), pp. 552-559.
[5] P.T. Johnstone, Stone spaces, Cambridge studies in Advanced Mathematics, Cambridge University press, cambridge, 1982.
[6] Y.M. Li, Exponentiable objects in the category of topological molecular lattices, Fuzzy sets and systems, 104 (1999), pp. 407-414.
[7] J. Picado and A. Pultr, Frames and locales, Topology Without Points, Frontiers in Mathematics, Birkhauser-Springer AG, Basel, 2012.
[8] S. Roman, Lattice and ordered sets, Springer, New York, 2008.
[9] W.J. Thron, Lattice-equivalence of topological spaces, Duke Math. J., 29 (1962), pp. 671-679.
[10] G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992), pp. 351-376.
[12] G.J. Wang, Generalized topological molecular lattices, Scientia Sinica, 8 (1984), pp. 785-798.