  In a partially ordered semigroup with the duality (or polarity) transform, it
is possible to define a generalisation of continued fractions. General
sufficient conditions for convergence of continued fractions with deterministic
terms are provided. Two particular applications concern the cases of convex
sets with the Minkowski addition and the polarity transform (where also
necessary and sufficient conditions of convergence for continued fractions with
constant terms are obtained) and the family of non-negative convex functions
with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.
