%0 Journal Article %T DIVISOR TOPOLOGIES AND THEIR ENUMERATION %J Journal of Algebraic Systems %I Shahrood University of Technology %Z 2345-5128 %A Esmaeeli, F. %A Mirzavaziri, K. %A Mirzavaziri, M. %D 2022 %\ 09/01/2022 %V 10 %N 1 %P 111-119 %! DIVISOR TOPOLOGIES AND THEIR ENUMERATION %K Topology %K Divisor topology %K Semi-divisor topology %R 10.22044/jas.2021.9712.1473 %X ‌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. %U https://jas.shahroodut.ac.ir/article_2325_51e352c41a02d11fd7a3511d42f35baf.pdf