On extensions and restrictions of τ-maxitive idempotent measures

In the paper we study τ-maxitive idempotent measures and consider their extensions and restrictions. For an idempotent measure we prove that its extension and restriction are τ-maxitive if and only if the given idempotent measure is τ-maxitive. Then we show that the composition of the extension operation and the restriction operation is an identity map.

FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS

Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.

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.

On central primitive idempotent measures

Let S be a compact semitopological semigroup and let P(S) be the convolution semigroup of probability measures on S. An idempotent measure μ in P(S) is defined to be primitive if and only idempotent measures in μP(S)μ are μ and the zero element m of P(S). In a previous paper [2] we give some characterization of primitive idempotent measures on S. Let Π(P(S)) be the set of primitive idempotents in P(S) and let Πc be the set of central primitive idempotents in P(S). It is shown in [1] that Π(P(S)) is neither an ideal nor even a subsemigroup of P(S) in general. The purpose of this paper is to investigate the structure of Πc.

Some problems on idempotent measures on semigroups

Essentially this paper does the following: In Section 2 it gives necessary and sufficient conditions in order that the support of an idempotent measure on a locally compact semigroup S, be completely simple. In Section 3 it proves that if I is an ideal of S of positive measure μ (= any probability measure), then μn (I) strictly increases to the limit 1. If in addition μ is idempotent, then μ (N-1N) and μ(NN-1) are positive for any open set N. In Section 4 certain compactness conditions are proven equivalent to joint weak*–continuity of the convolution of bounded measures and a limit theorem concerning the convolution powers (Cesarò sums) of μ is proven.