L-Convexity and Lattice-Valued Capacities

$L$-idempotent analogues of convexity are introduced ($L$ is a completely distributive lattice). It is proved that the category of algebras for the monad of $L$-valued capacities (regular plausibility measures) in the category of compacta is isomorphic to the category of $L$-idempotent biconvex compacta and their biaffine maps. For the functor of $L$-valued $\cup$-capacities ($L$-possibility measures) a family of monads parameterized by monoidal operations $*:L\times L\to L$ is introduced and it is shown that the category of algebras for each of these monads is isomorphic to the category of $(L,\oplus,*)$-convex compacta and their affine maps.