For the rest of the 'civilised' world, lowest fares only exist with consolidators and unpublished resources.
Airlines do not publish their negotiated fares (normally in economy class) on CRS' with the sole exception from SINGAPORE. Some airlines like CX, UA, SQ have fares which are loaded onto various CRS which can display their lowest special fares.
But for the vast majority of airlines including the ones above, CRS do not display special low negotiated fares. Hence the travel websites will find resistence by the general public to use them for online booking.