Compact affine monoids, harmonic analysis and information theory (Q2918877)

The authors revisit the theory of compact monoids and affine semigroups. The paper consists of three parts. In the first part, entitled ``Compact monoids and semigroups'', they prove some fundamental theorems about compact semigroups and monoids. The authors recall the notion of compact affine monoid and give three different proofs of the following fundamental result from the theory of compact affine monoids. A compact affine group \(G\) is a point.NEWLINENEWLINEThe second part entitled ``Compact affine monoids: probability measures, Wendel's theorem and the monoid stochastic matrices'' contains results about measures on compact semigroups and groups. If \(G\) is a compact semigroup, then the set \(P(G)\) of all probability measures endowed with a suitable topology and affine structure is a locally convex, affine, compact semigroup (Corollary 4.7). Using the properties of \(P(G)\) for a compact group, a proof is given of the existence of a Haar measure on a compact group. Another important example of a compact affine monoid is the set \(ST(n)\) of \(n\times n\) stochastic matrices.NEWLINENEWLINEThe goal of the third part is to generalize the results of [\textit{K. Martin} et al., Theor. Comput. Sci. 411, No. 19, 1918--1927 (2010; Zbl 1184.94201)] from binary channels to \(n\)-ary channels. The elements of \(ST(n)\) are interpreted as \(n\)-ary channels.NEWLINENEWLINEThe paper contains some historical notes about the development of the theory of compact semigroups.NEWLINENEWLINEFor the entire collection see [Zbl 1245.00037].