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Strictly speaking, operator== can be implemented in terms of operator< so that is not absolutely true, but if you intend to make use of the convenience provided using namespace std::rel_ops; then...

Personally, I would dispense with the ternary operator as I think that it would suffice to write:

bool operator< (const pthread_t& lhs, const pthread_t& rhs)
{
return memcmp(lhs.p, rhs.p,...

You don't seem to have overloaded operator< for your struct type in the first place, but yes, you can indeed use another comparator besides operator< if you desire as the second template parameter...