It is well-known that a
semigroup is subdirectly irreducible if and only if it has a minimum nontrivial
congruence. From this point of view, it is natural to call a semigroup right (left)
subdirectly irreducible if and only if it has a minimum nontrivial right (left)
congruence. It turns out that such semigroups are exactly the subdirectly irreducible
semigroups for which the minimum nontrivial congruence is also a minimum
nontrivial right (left) congruence. These semigroups form a class of subdirectly
irreducible semigroups for which results similar to those obtained by Schein for
commutative subdirectly irreducible semigroups are obtained. In fact, since a
commutative semigroup is subdirectly irreducible if and only if it is right
subdirectly irreducible, some of the results of this paper offer additional
knowledge on the structure of subdirectly irreducible semigroups of the third
kind.
