The genetic material, in the form of deoxyribonucleic acid (DNA), continues to shorten with each cell division, and cells eventually stop dividing when they sense that their DNA is critically shortened.
A somewhat similar concept is that of a clonal colony (also called a genet), wherein the cells (usually unicellular) also share a common ancestry, but which also requires the products of clonal expansion to reside at "one place", or in close proximity.
For instance, two plasma cells belonging to the same clone could be derived from different memory cells (in turn with shared clonality) and could be residing in quite distant locations, such as the cervical (in the neck) and inguinal (in the groin) lymph nodes.
The single-cell eukaryote Paramecium tetraurelia can undergo both asexual and sexual reproduction.
Binary fission involves mitosis-like behavior of the chromosomes similar to that of cells in higher organisms.