@article { author = {Esmaeeli, F. and Mirzavaziri, K. and Mirzavaziri, M.}, title = {DIVISOR TOPOLOGIES AND THEIR ENUMERATION}, journal = {Journal of Algebraic Systems}, volume = {10}, number = {1}, pages = {111-119}, year = {2022}, publisher = {Shahrood University of Technology}, issn = {2345-5128}, eissn = {2345-511X}, doi = {10.22044/jas.2021.9712.1473}, 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 = {Topology,Divisor topology,Semi-divisor topology}, url = {https://jas.shahroodut.ac.ir/article_2325.html}, eprint = {https://jas.shahroodut.ac.ir/article_2325_51e352c41a02d11fd7a3511d42f35baf.pdf} }