Fully simple semihypergroups, transitive digraphs, and sequence A000712

Fully simple semihypergroups have been introduced in [9], motivated by the study of the transitivity of the fundamental relation β in semihypergroups. Here, we determine a transversal of isomorphism classes of fully simple semihypergroups with a right absorbing element. The structure of that transversal can be described by means of certain transitive, acyclic digraphs. Moreover, we prove that, if n   is an integer ≥2, then the number of isomorphism classes of fully simple semihypergroups of size n+1n+1, with a right absorbing element, is the (n+1)(n+1)-th term of sequence A000712 in [20], namely, ∑k=0np(k)p(n−k), where p(k)p(k) denotes the number of nonincreasing partitions of integer k.