On covariety lattices

This paper shows basic properties of covariety lattices. Such lattices are shown to be infinitely distributive. The covariety lattice ${\displaystyle L_{CV}\left(K\right)}$ of subcovarieties of a covariety K of F-coalgebras, where F:Set → Set preserves arbitrary intersections is isomorphic to the lattice of subcoalgebras of a ${\displaystyle P_{\kappa }}$-coalgebra for some cardinal κ. A full description of the covariety lattice of Id-coalgebras is given. For any topology τ there exist a bounded functor F:Set → Set and a covariety K of F-coalgebras, such that ${\displaystyle L_{CV}\left(K\right)}$ is isomorphic to the lattice (τ,∪,∩) of open sets of τ.