## Abstract and Applied Analysis

### Lattice-Valued Convergence Spaces: Weaker Regularity and $p$-Regularity

#### Abstract

By using some lattice-valued Kowalsky’s dual diagonal conditions, some weaker regularities for Jäger’s generalized stratified $L$-convergence spaces and those for Boustique et al’s stratified $L$-convergence spaces are defined and studied. Here, the lattice $L$ is a complete Heyting algebra. Some characterizations and properties of weaker regularities are presented. For Jäger’s generalized stratified $L$-convergence spaces, a notion of closures of stratified $L$-filters is introduced and then a new $p$-regularity is defined. At last, the relationships between $p$-regularities and weaker regularities are established.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 328153, 11 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395858524

Digital Object Identifier
doi:10.1155/2014/328153

Mathematical Reviews number (MathSciNet)
MR3166598

Zentralblatt MATH identifier
1315.54007

#### Citation

Li, Lingqiang; Jin, Qiu. Lattice-Valued Convergence Spaces: Weaker Regularity and $p$ -Regularity. Abstr. Appl. Anal. 2014 (2014), Article ID 328153, 11 pages. doi:10.1155/2014/328153. https://projecteuclid.org/euclid.aaa/1395858524

#### References

• H. J. Kowalsky, “Limesräume und Komplettierung,” Mathematische Nachrichten, vol. 12, pp. 301–340, 1954.
• C. H. Cook and H. R. Fischer, “Regular convergence spaces,” Mathematische Annalen, vol. 174, no. 1, pp. 1–7, 1967.
• D. C. Kent and G. D. Richardson, “$p$-regular convergence spaces,” Mathematische Nachrichten, vol. 149, pp. 215–222, 1990.
• S. A. Wilde and D. C. Kent, “$p$-topological and $p$-regular: dual notions in convergence theory,” International Journal of Mathematics and Mathematical Sciences, vol. 22, pp. 1–12, 1999.
• W. Gähler, “Monadic topology–-a new concept of generalized topology,” in Recent Developments of General Topology, vol. 67 of Mathematical Research, pp. 136–149, Akademie, Berlin, Germany, 1992.
• D. C. Kent and G. D. Richardson, “Convergence spaces and diagonal conditions,” Topology and its Applications, vol. 70, no. 2-3, pp. 167–174, 1996.
• G. Jäger, “A category of $L$-fuzzy convergence spaces,” Quaestiones Mathematicae, vol. 24, pp. 501–517, 2001.
• J. M. Fang, “Stratified $L$-ordered convergence structures,” Fuzzy Sets and Systems, vol. 161, no. 16, pp. 2130–2149, 2010.
• J. M. Fang, “Relationships between $L$-ordered convergence structures and strong $L$-topologies,” Fuzzy Sets and Systems, vol. 161, no. 22, pp. 2923–2944, 2010.
• G. Jäger, “Subcategories of lattice-valued convergence spaces,” Fuzzy Sets and Systems, vol. 156, no. 1, pp. 1–24, 2005.
• G. Jäger, “Pretopological and topological lattice-valued convergence spaces,” Fuzzy Sets and Systems, vol. 158, no. 4, pp. 424–435, 2007.
• G. Jäger, “Fischer's diagonal condition for lattice-valued convergence spaces,” Quaestiones Mathematicae, vol. 31, no. 1, pp. 11–25, 2008.
• G. Jäger, “Lattice-valued convergence spaces and regularity,” Fuzzy Sets and Systems, vol. 159, no. 19, pp. 2488–2502, 2008.
• G. Jäger, “Gähler's neighbourhood condition for lattice-valued convergence spaces,” Fuzzy Sets and Systems, vol. 204, pp. 27–39, 2012.
• L. Li, Many-valued convergence, many-valued topology, and many-valued order structure [Ph.D. thesis], Sichuan University, 2008, (Chinese).
• L. Li and Q. Jin, “On adjunctions between Lim, S\emphL-Top, and S\emphL-Lim,” Fuzzy Sets and Systems, vol. 182, no. 1, pp. 66–78, 2011.
• L. Li and Q. Jin, “On stratified $L$-convergence spaces: pretopological axioms and diagonal axioms,” Fuzzy Sets and Systems, vol. 204, pp. 40–52, 2012.
• D. Orpen and G. Jäger, “Lattice-valued convergence spaces: extending the lattice context,” Fuzzy Sets and Systems, vol. 190, pp. 1–20, 2012.
• W. Yao, “On many-valued stratified $L$-fuzzy convergence spaces,” Fuzzy Sets and Systems, vol. 159, no. 19, pp. 2503–2519, 2008.
• H. Boustique and G. Richardson, “A note on regularity,” Fuzzy Sets and Systems, vol. 162, no. 1, pp. 64–66, 2011.
• H. Boustique and G. Richardson, “Regularity: lattice-valued Cauchy spaces,” Fuzzy Sets and Systems, vol. 190, pp. 94–104, 2012.
• P. V. Flores, R. N. Mohapatra, and G. Richardson, “Lattice-valued spaces: fuzzy convergence,” Fuzzy Sets and Systems, vol. 157, no. 20, pp. 2706–2714, 2006.
• P. V. Flores and G. Richardson, “Lattice-valued convergence: diagonal axioms,” Fuzzy Sets and Systems, vol. 159, no. 19, pp. 2520–2528, 2008.
• B. Losert, H. Boustique, and G. Richardson, “Modifications: lattice-valued structures,” Fuzzy Sets and Systems, vol. 210, pp. 54–62, 2013.
• L. Li and Q. Jin, “p-Topologicalness and p-regularity for lattice-valued convergence spaces,” Fuzzy Sets and Systems, 2013.
• R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic, New York, NY, USA, 2002.
• U. Höhle and A. Šostak, “Axiomatic foundations of fixed-basis fuzzy topology,” in Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, U. Höhle and S. E. Rodabaugh, Eds., vol. 3 of The Handbooks of Fuzzy Sets Series, pp. 123–273, Kluwer Academic, London, UK, 1999.
• G. Jäger, “Lowen fuzzy convergence spaces viewed as $L$-fuzzy convergence spaces,” The Journal of Fuzzy Mathematics, vol. 10, pp. 227–236, 2002.
• G. Preuss, Fundations of Topology, Kluwer Academic, London, UK, 2002.
• D. Zhang, “An enriched category approach to many valued topology,” Fuzzy Sets and Systems, vol. 158, no. 4, pp. 349–366, 2007.
• G. Jäger, “Diagonal conditions for lattice-valued uniform convergence spaces,” Fuzzy Sets and Systems, Fuzzy Sets and Systems, vol. 210, pp. 39–53, 2013. \endinput