Abstract
A kind of relational mappings, generated from usual set-valued mappings by adding some relative equivalences, is introduced in this paper. With this interesting and powerful tool, one can conveniently discuss some classes of orthogroups and some more general classes of semigroups in the scope of semisuperabundant semigroups. Relational homomorphisms over some congruences on semigroups are used to clarify the definition of a refined semilattice of semigroups. With this notion we investigate the properties and describe the structure of regular orthocryptou semigroups. Such description can also be specialized to various subclasses of regular orthocryptou semigroups, such as left quasi-normal orthocryptou semigroups, left regular orthocryptou semigroups, left C-rpp semigroups in which \widetilde{H} is a congruence, regular orthocryptogroups, left quasi-normal orthocryptogroups, left regular orthocryptogroups, Clifford semigroups, regular bands, left quasi-normal bands and left regular bands.
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