LAUSR

A rectangular group is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. Some special cases of rectangular groups consisting of left groups and right groups are also considered here. Let $$ \mathrm {Cay}(S,A) $$Cay(S,A) denote the Cayley digraph of the rectangular group S with the connection set A. In this paper, we are interested in studying some properties of $$ \mathrm {Cay}(S,A) $$Cay(S,A) such as the independence, weakly independence, path independence, and weakly path independence. Furthermore, the independence numbers for those properties of $$ \mathrm {Cay}(S,A) $$Cay(S,A) are also determined.