Inferior semigroups, left (right) inferior ideals and inferior quasi-ideals in semi-groups are introduced, and several properties are investigated. Characterizations of inferior semigroups and ideals are considered, and relations between inferior semigroups, inferior ideals and inferior quasi-ideals are discussed. Characteristic inferior mappings and inferior products of inferior mappings are introduced. Using these notions, related properties on inferior semigroups, left (right) inferior ideals and inferior quasi-ideals are investigated. A regular semigroup is characterized by an inferior quasi-ideal.
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