  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 $P, P_0, R_0$
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-$P_0$ and $R_0$, or the $P_0$ and $R_0$
properties then all the associated HCCPs have solutions. In particular, if a
continuous 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). We
present a necessary condition for a nonlinear transformation to have the GUS
property. Moreover, we establish a global error bound for the HCCP with the
uniform-trace-$P$ property. Finally, we study the HCCP with the relaxation
transformation on a $T$-algebra and automorphism invariant properties for
homogeneous cone linear complementarity problem.
