Presenting the frame of the unit circle

We present the frame L(T)L(T) of the unit circle by generators and relations in two alternative ways. The first is the localic counterpart of the Alexandroff compactification of the real line while the other can be understood as a localic analogue of the quotient space R/ZR/Z. With an eye towards a prospective point-free description of Pontryagin duality, we then show how the usual group operations of the frame of reals can be lifted to the new frame L(T)L(T), endowing it with a canonical localic group structure.