ABSTRACT:Lawson has given a sufficient condition for a semigroup S which guarantees that S does not admit a ring structure. From Lawson's theorem, we have that the multiplicative interval semigroup [0, 1] does not admit a ring structure. In this paper we give an elementary proof of this fact. We then show that the multiplicative interval semigroup [a, 1] with −1 a < 0 < a 2 1 does not admit the structure of a ring, a fact which cannot be derived from Lawson's theorem. These facts are then applied to show that every nontrivial multiplicative bounded interval semigroup on R does not admit a ring structure.
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