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Original Articles

Strong identities and fortification in transposition hypergroups

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Pages 169-193 | Published online: 03 Jun 2013
 

Abstract

An element e in a hypergroup H is a strong identity if x ∈ ex = xexe. The elements of H separate into two classes, the set A = {x ∈ H | ex = xe = xe}, 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” Ce, which is a polygroup under the relativized hyperoperation. Therefore, H can be reconstructed from A and Ce, as H is isomorphic to the expansion of the polygroup Ce 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.

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