Abstract
The class of all regular semigroups with inverse transversals is closed under direct products, IST-subsemigroups and homomorphic images. It forms an IST-variety \(\mathcal{IST}\). The IST-variety is different from not only the regular unary semigroup varieties, but also the regular semigroup e-varieties. \(\mathcal{IST}\) contains many IST-varieties as its subvarieties. For example, it contains the IST-variety of all orthodox semigroups with inverse transversals \(\mathcal{OIST}\), the IST-variety of all (left regular, right regular) bands with inverse transversals (\(\mathcal{LRBIT}\), \(\mathcal{RRBIT}\)) \(\mathcal{BIT}\), the IST-variety of all regular semigroups with Q-inverse transversals \(\mathcal{QIST}\), etc. According to the descriptions for the free objects in \(\mathcal{LRBIT}\), \(\mathcal{RRBIT}\) and \(\mathcal{BIT}\), we characterize further in this paper the words in the following free IST-semigroups: the free IST-bands in \(\mathcal{BIT}\), the free left regular IST-bands in \(\mathcal{LRBIT}\) and the free right regular IST-bands in \(\mathcal{RRBIT}\) respectively. Classifications of Blyth for regular semigroups with inverse transversals and characterizations of Gerhardt for words on free bands are hence generalized and enriched.
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