Equilogical spaces and algebras for a double-power monad.

G. Frosoni, G. Rosolini

We investigate the algebras for the double-power monad on the Sierpisnki space in the category Equ of equilogical spaces, a cartesian closed extension of Top0 introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of Equ. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of Equ.
We also analyse the case of algebras in the category Top0 of T0-spaces, again seen as a full subcategoy of Equ, proving that each algebra for the double-power monad in Top0 has an underlying sober, compact, connected space. 

Tbilisi Mathematical Journal, Vol. 10(3) (2017), pp. 121-139