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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 .

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