Every semigroup containing an ideal subgroup is called a homogroup. Let $ (S,\cdot)$ be homomgroup, hence it contains an ideal $ I$ that is also a subgroup. It is easy to see that $ I$ is the least ideal, a maximal subgroup of $ S$ , and its identity (denoted by $ e_I$ ) is a central idempotent of $ S$ . Now,
(1) Is the ideal subgroup of $ S$ unique?
(2) Is $ e_I$ the only central idempotent of $ S$ ?
(3) Is $ I$ the largest subgroup of $ S$ ?
(4) Is $ e_I$ the identity element of the subgroup of all central idempotents of $ S$ ?
(5) Is $ e_I$ a zero element of the set of all idempotents of $ S$
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