In this paper we investigate important categories lying strictly between the Kleisli category and the Eilenberg-Moore category, for a Kock-Z\"oberlein monad on an order-enriched category. Firstly, we give a characterisation of free algebras in the spirit of domain theory. Secondly, we study the existence of weighted (co)limits, both on the abstract level and for specific categories of domain theory like the category of algebraic lattices. Finally, we apply these results to give a description of the idempotent split completion of the Kleisli category of the filter monad on the category of topological spaces.Comment: small corrections

@article{HofmannS17,
  title = {Aspects of algebraic Algebras},
  author = {Dirk Hofmann and Lurdes Sousa},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 3},
  year = {2017},
  doi = {10.23638/lmcs-13(3:4)2017}
}