Some Consistent Power Constructions

Consistent Hoare, Smyth and Plotkin power domains are introduced and discussed by Yuan and Kou. The consistent algebraic operation $+$ defined by them is a binary partial Scott continuous operation satisfying the requirement: $a+b$ exists whenever there exists a $c$ which is greater than $a$ and $b$. We extend the consistency to be a categorical concept and obtain an approach to generating consistent monads from monads on dcpos whose images equipped with some algebraic operations. Then we provide two new power constructions over domains: the consistent Plotkin index power domain and the consistent probabilistic power domain. Moreover, we verify these power constructions are free.