The centralizer of an element in a reversible hypergroup
Let \(H\) be a hypergroup and \(I(H)\) be the set of its identities (an element \(e \in H\) is an identity of \(H\) if \(e \in ex \cap xe\), for every \(x \in H)\). If \(x \in H\), denote by \(i(x) = \{y \in H \mid xy \cap yx \cap I(H) \neq \emptyset\} =\) the set of the inverses of \(x\) and by \({\mathcal C}(x)\) the least complete part of \(H\) containing \(x\). We also consider \(A(x) = \{y \in H \mid {\mathcal C}(xy) \cap {\mathcal C}(yx) \neq \emptyset\}\) and \({\mathcal C}_H(x) = A(x) \cup A^2(x) \cup A^3(x) \cup \dots\). The hypergroup \(H\) is regular if \(I(H) \neq \emptyset\) and \(i(x) \neq \emptyset\), for every \(x \in H\). If, more than that, for every \(x, y, z\) in \(H\) satisfying \(x \in yz\) there exist \(y' \in i(y)\) and \(z' \in i(z)\) such that \(z \in y' x\) and \(y \in xz'\) then \(H\) is reversible. The aim of this paper is to introduce a notion of centralizer for regular and reversible hypergroups and to establish some of its properties. Thus it is shown that, for every \(x \in H\), the set \({\mathcal C}_H(x)\) is a superhypergroup of \(H\), called the centralizer of \(x\) in \(H\). Particularly, for complete hypergroups, whence for groups, too, the centralizer of an element \(x\) is \({\mathcal C}(x) = \{y \in H \mid xy = yx\}\) so it coincides with the classical one. It is also established that in a regular and reversible hypergroup \(H\) the following conditions hold, for every \(x\) in \(H\): (i) \(({\mathcal C}(x) \cup \omega_H) \subset {\mathcal C}_H(x)\); (ii) \({\mathcal C}_H(x) = {\mathcal C}_H(x')\), for every \(x' \in i(x)\); (iii) \({\mathcal C}_H(x) = H\) if \(x \in \omega_H\); (iv) \({\mathcal C}_H (x) = {\mathcal C}_H (y)\), if \(xy \cap \omega_H \neq \emptyset\).
scientific article; zbMATH DE number 790784