Document Type : Original Manuscript

Authors

1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, Iran.

2 Department of Computer Science, School of Mathematics, Statistics and Computer Science, University of Tehran, P.O. Box 141556619, Tehran, Iran.

Abstract

‎For a positive integer $m$‎, ‎a subset of divisors of $m$ is called a \textit{divisor topology on $m$} if it contains $1$ and $m$ and it is closed under taking $\gcd$ and $\rm lcm$‎. ‎If $m=p_1\dots p_n$ is a square free positive integer‎, ‎then a divisor topology $m$ corresponds to a topology on the set $[n]=\{1,2,\ldots,n\}$‎. ‎Giving some facts about divisor topologies‎, ‎we give a recursive formula for the number of divisor topologies on a positive integer‎.

Keywords

###### ##### References
1. T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, 1976.
2. M. Benoumhani, The number of topologies on a finite set, J. Integer Seq., 9 (2006), Article ID: 06.2.6, 9 pp.
3. J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2n (except for some special cases), Discr. Math., 154 (1996), 27–39.
4. T. Clark and T. Richmond, The number of convex topologies on a finite totally ordered set, Involve, 8(1) (2015), 25–32.
5. M. Erné, Struktur- und anzahlformeln fuer topologien auf endlichen mengen, Manuscripta Math., 11 (1974), 221–259.
6. M. Erné and K. Stege, Counting finite posets and topologies, Order,
8 (1991), 247–265.
7. J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295–297.
8. F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.
9. J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order, 17(4) (2000), 333–341.
10. D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc., 25 (1970), 276–282.
11. M. Kolli, Direct and elementary approach to enumerate topologies on a finite set, J. Integer Seq., 10 (2007), Article ID: 07.3.1, 11 pp.
12. M. Kolli, On the cardinality of the T0-topologies on a finite set, Int. J. Comb., (2014), Article ID: 798074, 7 pp.
13. R. E. Larson and S. J. Andima, The lattice of topologies: A survey equations, Rocky Mountain J. Math., 5(2) (1975), 177–198.
14. H. Levinson, R. Silverman, Topologies on finite sets, II. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 699–712, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561090 (81c:54006) - From N. J. A. Sloane, Jun 05 2012.
15. M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math. Soc., 19 (1968), 885–889.
16. A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194–198.
17. N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Sequence A000798, Academic Press, 1995.