The orbits of generalized orthogonal monoids under the action of their unit groups

Abstract

We first introduce a class of new monoids: generalized orthogonal monoids. These monoids are examples of dilating monoids and contain the traditional orthogonal monoids as proper submonoids. We then investigate their orbit structures under the two-sided action of their unit groups. We next determine the complete set of all the orbits of these monoids M by introducing the concept of standard generalized orthogonal matrices. Note that the set of all standard generalized orthogonal matrices is not a submonoid of M. We further describe the orbits in M by introducing the set Y of standard generalized orthogonal matrices of the second kind. We show that Y is a submonoid of M, and we determine the partial order on Y and the corresponding Hasse diagram. We also compute the total number of the orbits in M.