Hypergroups of type \(C\)
A hypergroup \((H,\circ)\) verifying the following two conditions: i) there exists \(e\in H\) such that \(x\circ e=x\), for any \(x\in H\) (\(e\) is the (right) identity of \(H\)); ii) for any \(x,y,z\) in \(H\), \(x\circ y\cap x\circ z\neq\emptyset\) implies \(e\circ y=e\circ z\); is said to be a hypergroup of type \(C\) (at the right) ((right) weak cogroup). If moreover \((H,\circ)\) verifies the condition: iii) \(\text{Card}(x\circ y)=\text{Card}(e\circ y)\), for any \(x,y\) in \(H\); then \((H,\circ)\) is called a (right) cogroup. Any \(D\)-hypergroup, \((G/{\mathcal G},\circ)\), where \(G\) is a group and \(\mathcal G\) is a (not necessarily normal) subgroup of \(G\), is a cogroup[\textit{J. Eaton}, Duke Math. J. 6, 101-107 (1940; Zbl 0027.00804)] but there exist cogroups which are not \(D\)-hypergroups[\textit{Y. Utumi}, Osaka Math. J. 1, 73-80 (1949; Zbl 0036.293)]. In this paper the most important properties of the hypergroups of type \(C\) and many examples which situate these hypergroups among the classes of hypergroups already known are presented. The morphisms and the quotient hypergroups of the hypergroups of type \(C\) are studied. Among the main results established in the paper, the following theorem presents a construction which is analogous to the semidirect product of groups: Theorem. Let \((H,\circ)\) be a cogroup and let \(e\) be the right identity of \(H\). Consider \(\mathcal G\) a group of permutations of \(H\) which let invariant the partition \(P=\{e\circ x\mid x\in H\}\) of \(H\) and \((\phi_ x)_{x\in H}\) a family of surjective maps \(\phi_ x: H\to H\), for any \(x\in H\). Assume that: i) \(\phi_ x(e\circ y)=e\circ y\), for any \(x,y\) in \(H\); ii) \(\phi_ e=1_ H\); iii) \(\phi_ x(\sigma(\phi_ y(\tau(z))))=\phi_{\phi_ x(\sigma(y))}((\sigma\circ\tau)(z))\), for any \(\sigma,\tau\) in \(\mathcal G\) and \(x,y,z\) in \(H\); Define on \(G=H\times {\mathcal G}\) the operation ``\(\cdot\)'' by: \((x,\sigma)\cdot (y,\tau)=(\phi_ x(\sigma(y)),\sigma\circ\tau)\). Then \((G,\cdot)\) is a group, \(K=\{e\}\times {\mathcal G}\) is a subgroup of \(G\) and \((H,\cdot)\) is isomorphic to the \(D\)-hypergroup \(G/K\). As application, all the hypergroups of type \(C\) having at most five elements (up to isomorphism) are determined and it is shown that all of them are \(D\)-hypergroups. For other results about cogroups and weak cogroups see \textit{L. Haddad}, \textit{Y. Sureau} [Pac. J. Math. 145, 17-58 (1990; Zbl 0663.20070)].
scientific article; zbMATH DE number 60499
scientific article