Free product on semihypergroups (Q2227618)

A non-empty locally compact Hausdorff space \((K,\ast)\) is called a (topological) semihypergroup if the following conditions are satisfied: \begin{itemize} \item[(i)] \((M(K),+,\ast)\) is an associative algebra, where \(M(K)\) is the space of all regular complex Borel measures on \(K\). \item[(ii)] The bilinear mapping \(\ast:M(K)\times M(K)\to M(K)\) is positive continuous. \item[(iii)] For all \(x,y\in K\), \(p_x \ast p_y\) is a probability measure with compact support. \item[(iv)] The mapping \((x,y)\mapsto supp(p_x \ast p_y)\) from \(K\times K\) into \(\mathcal{C}(K)\) is continuous, where \(\mathcal{C}(K)\) is the space of compact subsets of \(K\) endowed with the Michael topology, that is the topology generated by the subbasis of all \(\mathcal{C}_U (V)=\{C\in \mathcal{C}(K)\mid C\cap U\not=\emptyset,\ C\subset V\}\) for which \(U\) and \(V\) are open subsets of \(K\). \end{itemize} In this paper, the author continues her studies on semihypergroups which initiated in [Semigroup Forum 100, 671--697 (2020; Zbl 1472.20148)], and discusses how a semihypergroup homomorphism naturally induces a homomorphism between measure algebras. It is also discussed how free products can be defined on semihypergroups, abiding by the general norms of defining a free product given any category of objects in general. Moreover, a subclass of pure non-trivial semihypergroups is introduced and the existence of a unique free product is investigated given such a family of objects. Furthermore, it is shown that the natural free product structure along with the naturally induced topology, which fails to give a useful free product structure for topological groups, indeed works well here and gives a natural free product structure for a class of non-trivial semihypergroups. As expected, the existence of a natural free product ensures the abundance of new examples and opens up several new paths of research including compactification of a given semihypergroup and amalgamation of different kinds of semihypergroups in pursuit of specific examples and counter-examples among others.