In this article, by considering some presentations of groups and semigroups, we investigate the structure of the groups and semigroups presented by them and introduce some infinite classes of semigroups which have kite-like egg-boxes of Dclasses. These semigroups have a unique idempotent and the minimal two-sided ideal of them is isomorphic to the group presented by the same presentation as for these semigroups. All of the Green's relations in these semigroups coincide and every proper subsemigroup of them is a subgroup.
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