On Extension Of Functors

A.Chigogidze defined for each normal functor on the category Comp an extension which is a normal functor on the category Tych. We consider this extension for any functor on the category Comp and investigate which properties it preserves from the definition it preserves from the definition of normal functor. We investigate as well some topological properties of such extension

Опуклості, породжені монадами

Let F be a monad in the category Comp. We build a convexity in general sense for each F-algebra (see [1]), investigate properties of such convexities, and apply them to prove that the multiplication map of the monad of order-preserving functionals is soft

Invariant idempotent measures

The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the notion of probability measure. The idempotent measures found numerous applications in mathematics and related areas, in particular,  the optimization theory, mathematical morphology, and game theory. In this note we introduce the notion of invariant idempotent measure for an iterated function system in a complete metric space. This is an idempotent counterpart of the notion of invariant probability measure defined by Hutchinson. Remark that the notion of invariant idempotent measure was previously considered by the authors for the class of ultrametric spaces. One of the main results is the existence and uniqueness theorem for the invariant idempotent measures in complete metric spaces. Unlikely to the corresponding Hutchinson's result for invariant probability measures, our proof does not rely on metrization of the space of idempotent measures. An analogous result can be also proved for the so-called in-homogeneous idempotent measures in complete metric spaces. Also, our considerations can be extended to the case of the max-min measures in complete metric spaces