We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem (HCCP). Employing the T-algebraic characterization of homogeneous cones, we generalize the P, P0, R0 properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of HCCP. We prove that if a continuous function has either the order-P0 and R0, or the P0 and R0 properties then all the associated HCCPs have solutions. In particular, if a continuous monotone function has the trace-P property then the associated HCCP has a unique solution (if any); if it has the uniform-trace-P property then the associated HCCP has the global uniqueness (of the solution) property (GUS). Moreover, we establish a global error bound for the HCCP with the uniform-trace-P property under some conditions for homogeneous cone linear complementarity problem
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