Strong identities and fortification in transposition hypergroups

Abstract

An element e in a hypergroup H is a strong identity if x ∈ ex = xe ⊂ x ⋃ e. The elements of H separate into two classes, the set A = {x ∈ H | ex = xe = x ⋃ e}, including e, of attractive elements and the set C = {x ∈ H – e | ex = xe = x} of canonical elements. If H is a transposition hypergroup then A is shown to be a closed subhypergroup of essentially indistinguishable elements. The structure of H is then determined, for A can be contracted into e leaving the “resulting” C ⋃ e, which is a polygroup under the relativized hyperoperation. Therefore, H can be reconstructed from A and C ⋃ e, as H is isomorphic to the expansion of the polygroup C ⋃ e by the transposition hypergroup A through e. The study of transposition hypergroups containing a strong identity separates into the study of polygroups and the study of transposition hypergroups of all attractive elements. A fortified transposition hypergroup H is defined and shown to contain a unique strong identity. Moreover, each nonidentity element is shown to have unique nonidentity left and right inverses that are identical. For H consisting of all attractive elements, the subhypergroups K that are symmetric, K = K –1, are studied. The double cosets of K, the sets K \\ (x/K) = (K \\ x)/K if x ∉ K, otherwise K, partition H. The resulting quotient space H : K of double cosets is proven to be a fortified transposition hypergroup in which K is the strong identity and every member is attractive.