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Journal of Convex Analysis 21 (2014), No. 1, 029--052
Copyright Heldermann Verlag 2014

L-Convexity and Lattice-Valued Capacities

Oleh Nykyforchyn
Dept. of Mathematics and Computer Science, Vasyl' Stefanyk Precarpathian National University, Shevchenka 57, Ivano-Frankivsk 76025, Ukraine

Dusan Repovs
Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia


$L$-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.

Keywords: Capacity functor, algebra for a monad, idempotent semimodule, idempotent convexity, plausibility measure.

MSC: 18B30, 18C20, 06B35, 52A01

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