Arithmetic grids are a new instance in distributed and parallel computing systems. They can realize a virtual supercomputer over idle resources available in a wide area network like the Internet. The grids are characterized for exploiting highly heterogeneous resources. They focus on arithmetic grids to reach high performance by using effective map tasks onto heterogeneous resources. Generally, the mapping tasks are realized by online and batch methods. In the batch mode at any mapping event a batch of tasks are mapped, whereas in online mode only one task is mapped. In this paper, four on-line mode mapping algorithms based on learning automata are introduced. To show the effectiveness of the proposed algorithms, computer simulation has been conducted. The results of experiments show that the proposed algorithms outperform two best existing mapping algorithms when machine heterogeneity high.