Abstract
In this paper, several equivalent conditions for a monoid to be in the variety generated by finite (left, right) locally trivial monoids are given. The structure of a regular language, whose syntactic monoid without containing the congrence class of the empty word is a left locally, or right locally, or locally trivial semigroup, is completely determined. A description of *-variety of languages corresponding to the variety generated by finite locally trivial monoids is also given. Meanwhile, we also discuss the connections among the varieties generated by finite nilpotent monoids, left locally, right locally, and locally trivial monoids.
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