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Let $$S$$ be a semigroup and ' $$*$$ ' a unary operation on $$S$$ which satisfies the following identities Then, $$S^*=\{x^*|x\in S\}$$ is called a regular $$*$$ - transversal of $$S$$ in the literatures. Following Munn and Hall's idea, in this paper we construct fundamental regular semigroups with quasi-ideal regular $$*$$ -transversals by which fundamental representations of regular semigroups with quasi-ideal regular $$*$$ -transversals are obtained. [ABSTRACT FROM AUTHOR]