  In our previous work, we introduced McKinsey-Tarski algebras (MT-algebras for
short) as an alternative pointfree approach to topology. Here we study local
compactness in MT-algebras. We establish the Hofmann-Mislove theorem for sober
MT-algebras, using which we develop the MT-algebra versions of such well-known
dualities in pointfree topology as Hofmann-Lawson, Isbell, and Stone dualities.
This yields a new perspective on these classic results.
