It seems that posits have a value called NaR ("not a real") which serves the purpose of all the special values (including both infinities) in IEEE floats. There are some key differences though:

  NaR (posits)                 NaN (floats)
  ------------                 ------------
  one bit pattern              many different bit patterns
  always equal to itself       never equals itself
  less than all other values   falsifies all orderings