Extending Algebraic Operations to D-Completions

In this article we show how separately continuous algebraic operations on T0-spaces and the laws that they satisfy, both identities and inequalities, can be extended to the D-completion, that is, the universal monotone convergence space completion. Indeed we show that the operations can be extended to the lattice of closed sets, but in this case it is only the linear identities that admit extension. Via the Scott topology, the theory is shown to be applicable to dcpo-completions of posets. We also explore connections with the construction of free algebras in the context of monotone convergence spaces.