By Prof. Dr. John N. Mordeson, Prof. Davender S. Malik, Prof. Nobuaki Kuroki (auth.)

ISBN-10: 3540371257

ISBN-13: 9783540371250

ISBN-10: 3642057063

ISBN-13: 9783642057069

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A nonempty subset A of S is called a left (right) ideal of S if SA ~ A (AS ~ A). Further, A is called a two-sided ideal of S if it is both a left and a right ideal of S. A nonempty subset A of S is called an interior ideal of S if SAS ~ A, and a quasi-ideal of S if AS n SA ~ A. A subsemigroup A of Sis called a bi-ideal of S if ASA ~ A. A nonempty subset A is called a generalized bi-ideal of S if ASA ~ A. 1 A semigroup S is called regular if for each element a of S, there exists an element x E S such that a = axa.

Then f is a fuzzy bi-ideal of 8 and is not a fuzzy quasi-ideal of 8. 4 Let A be a nonempty subset of a semigroup 8. Then A is a quasi-ideal of 8 if and only if the ehameteristie funetion CAis a fuzzy quasi-ideal of 8. Proof. First assume that A is a quasi-ideal of 8. Let a be any element of 8. If a E A, then If a ~ A, then CA (a) = O. On the other hand, assume that ((CA 08) n (8 o CA)) (a) = 1. Then Va=pq{CA(P) Â 8(q)} = (CA o 8)(a) = 1 and This implies that there exist elements b, e, d and e of S with a = be = de such that and Hence a = bc = de E A8 n SA ~ A, 50 2.

1) Let f and 9 be two fuzzy subsemigrvups of S. Then f n 9 is also a fuzzy subsemigroup of S. (2) Let f and 9 be fuzzy lefi (right, two-sided) ideal of S. Then f n 9 is also a fuzzy lefi (right, two-sided) ideal of S. 42 2. Fuzzy Ideals Proof. (1) Let f and g be any fuzzy subsemigroups of S. Let a and b be any elements of S. Then (f n g)(ab) > > f(ab) 1\ g(ab) (f(a) 1\ f(b)) 1\ (g(a) 1\ g(b)) (f(a) 1\ g(a)) 1\ (f(b) 1\ g(b)) (f n g)(a) 1\ (f n g)(b). Thus f n g is a fuzzy subsemigroup of S. Property (2) can be proved in a similar manner.

### Fuzzy Semigroups by Prof. Dr. John N. Mordeson, Prof. Davender S. Malik, Prof. Nobuaki Kuroki (auth.)

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