L-Convexity and Lattice-Valued Capacities
Abstract
<jats:p> <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -idempotent analogues of convexity are introduced ( <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is a completely distributive lattice). It is proved that the category of algebras for the monad of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -valued capacities (regular plausibility measures) in the category of compacta is isomorphic to the category of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -idempotent biconvex compacta and their biaffine maps. For the functor of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -valued <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\cup</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∪</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -capacities ( <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -possibility measures) a family of monads parameterized by monoidal operations <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">*:L\times L\to L</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>:</mml:mo> <mml:mi>L</mml:mi> <mml:mo>×</mml:mo> <mml:mi>L</mml:mi> <mml:mo>→</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is introduced and it is shown that the category of algebras for each of these monads is isomorphic to the category of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">(L,\oplus,*)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mo>⊕</mml:mo> <mml:mo separator="true">,</mml:mo> <mml:mo>∗</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -convex compacta and their affine maps. </jats:p>