Fully fuzzy prime semigroups

The fundamental concept of a fuzzy set, introduced by Zadeh in his classic paper [5] of 1965, has been applied by many authors to generalize some of the basic notions of algebra. In this note we characterize the semigroups for which each fuzzy ideal is prime, also the semigroups for which each fuzzy right ideal is prime. In Section 2, we prove that a semigroup is fully fuzzy prime if and only if it is semisimple and its set of fuzzy ideals is totally ordered. In Section 3, we define the fuzzy prime right ideals of a semigroup and we prove that if the set of all fuzzy right ideals of S is totally ordered, then S is right weakly regular if and only if every fuzzy right ideal of S is a fuzzy prime right ideal. By a semigroup (S,·), we mean a nonempty set S together with an associative binary operation “·”. A semigroup S is called commutative if “·” is commutative, that is, a · b = b · a for all a,b ∈ S. A semigroup S is called a monoid if it has an identity element with respect to “·”. If S has no identity element, then it is easy to adjoin an identity element 1 to the set by defining 1 · s = s · 1 = s, for all s in S. We will use the notation S1 with the following meaning: