L-Convexity and Lattice-Valued Capacities

-idempotent analogues of convexity are introduced (

L L

is a completely distributive lattice). It is proved that the category of algebras for the monad of

L L

-valued capacities (regular plausibility measures) in the category of compacta is isomorphic to the category of

L L

-idempotent biconvex compacta and their biaffine maps. For the functor of

L L

-valued

\cup ∪

-capacities (

L L

-possibility measures) a family of monads parameterized by monoidal operations

*:L\times L\to L ∗ : L × L → 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,*) ( L , ⊕ , ∗ )

-convex compacta and their affine maps.