Based on a completely distributive lattice $L$, degrees of compatible $L$-subsets and compatible mappings are introduced in an $L$-approximation space and their characterizations are given by four kinds of cut sets of $L$-subsets and $L$-equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$-subsets and compatible degrees of $L$-subsets. Finally, the notion of complete $L$-sublattices is introduced and it is shown that the...

Let $\alpha$ be an infinite cardinal. Let ${𝒯}_{\alpha }$ be the class of all lattices which are conditionally $\alpha$-complete and infinitely distributive. We denote by ${𝒯}_{\sigma }^{\text{'}}$ the class of all lattices $X$ such that $X$ is infinitely distributive, $\sigma$-complete and has the least element. In this paper we deal with direct factors of lattices belonging to ${𝒯}_{\alpha }$. As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class ${𝒯}_{\sigma }^{\text{'}}$.