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Orthodox Transversals of Regular Semigroups

Orthodox transversals were introduced by the first author as a generalization of inverse transversals [Comm. Algebra 27(9) (1999), pp. 4275–4288]. One of our aims in this note is to consider the general case of orthodox transversals. The main results are on the sets I and Λ, two components of regular semigroups with orthodox transversals. We prove that the semibands Ī and generated by I and Λ respectively are bands. Also somewhat interesting generalizations of properties on orthodox semigroups are given; another aim in this note is to give some examples illustrating the situations that the class of regular semigroups with orthodox transversals properly includes the class of regular semigroups with inverse transversals as well as the class of orthodox semigroups.