Information Sciences 177 (2007) 5798–5800

www.elsevier.com/locate/ins

Short Communication

Note on ‘‘Generalized fuzzy interior ideals in semigroups’’

Yunqiang Yin, Hongxing Li *

School of Mathematical Sciences, Beijing Normal University, 100875 Beijing, China

Received 21 December 2006; received in revised form 11 March 2007; accepted 14 March 2007

Abstract

In this paper, we give a negative answer to an open question about (2, 2 _q)-fuzzy interior ideals, proposed in Jun andSong [Y.B. Jun, S.Z. Song, Generalized fuzzy interior ideals in semigroups, Inform. Sci. 176 (2006) 3079–3093].� 2007 Published by Elsevier Inc.

Keywords: Belong to; Quasi-coincident with; (2, 2 _q)-fuzzy interior ideals

1. Introduction

Jun and Song [1] introduced the concept of (2, 2 _q)-fuzzy interior ideals of a semigroup and proposed anopen question as follows. Let fAi j i 2 Kg be a family of (2, 2 _q)-fuzzy interior ideals of a semigroup S. IsA :¼ [i2KAi an (2, 2 _q)-fuzzy interior ideal of S?

The main aim of this paper is to solve the above open question. We give a negative answer to the openquestion by a counterexample and present a sufficient condition so that the question can be answered posi-tively. Moreover, we show by a counterexample that the proof of the necessity part of Theorem 4.12 in [1]is incorrect and give a new proof. For these purpose, we first recall the definition and theorem that will beused in this note.

Definition 1.1. ([1]) A fuzzy set A in a semigroup S is called an (2, 2 _q)-fuzzy interior ideal of S if it satisfiesthe following conditions:

(i) (8x; y 2 S) (AðxyÞP minfAðxÞ;AðyÞ; 0:5g),(ii) (8x; a; y 2 S) (AðxayÞP minfAðaÞ; 0:5g).

Theorem 1.2. ([1], Theorem 4.12) A fuzzy set A in S is an (2, 2 _q)-fuzzy interior ideal of S if and only if [A]t is

an interior ideal of S for all t 2 ð0; 1�, where ½A�t ¼ fx 2 S j xt 2 _qAg.

0020-0255/$ - see front matter � 2007 Published by Elsevier Inc.

doi:10.1016/j.ins.2007.03.016

* Corresponding author. Tel.: +86 10 58808787; fax: +86 10 58807482.E-mail addresses: lhxqx@bnu.edu.cn (Y. Yin), yinyunqiang@mail.bnu.edu.cn (H. Li).

Y. Yin, H. Li / Information Sciences 177 (2007) 5798–5800 5799

2. A counterexample to the open question

The following example gives a negative answer to the above question.

Counterexample 2.1. Let S ¼ fa; b; c; dg be a semigroup with the following multiplication table:

Let A1 and A2 be fuzzy subsets of S such that

A1ðaÞ ¼ 0:4; A1ðbÞ ¼ 0:4; A1ðcÞ ¼ 0; A1ðdÞ ¼ 0;

A2ðaÞ ¼ 0:4; A2ðbÞ ¼ 0; A2ðcÞ ¼ 0:4; A2ðdÞ ¼ 0:

Then both A1 and A2 are (2, 2 _q)-fuzzy interior ideals of S, but A1 [ A2 is not an (2, 2 _q)-fuzzy interiorideal of S, since 0 ¼ A1ðdÞ _ A2ðdÞ ¼ ðA1 [ A2ÞðdÞ ¼ ðA1 [ A2ÞðbcÞ < minfðA1 [ A2ÞðbÞ; ðA1 [ A2ÞðcÞ; 0:5g ¼minf0:4; 0:4; 0:5g ¼ 0:4.

The following theorem can be obtained if we present a sufficient condition.

Theorem 2.2. Let fAi j i 2 Kg be a family of (2, 2 _q)-fuzzy interior ideals of S such that Ai � Aj or Aj � Ai for

all i; j 2 K. Then A ¼ [i2KAi is an (2, 2 _q)-fuzzy interior ideal of S.

Proof

(1) For all x; y 2 S, we have

AðxyÞ ¼ ð[i2KAiÞðxyÞ ¼ _i2KAiðxyÞP _i2K minfAiðxÞ;AiðyÞ; 0:5gðsince Ai is an ð2;2 _qÞ-fuzzy interior ideal of S for all i 2 KÞ¼ minf_i2KAiðxÞ;_i2KAiðyÞ; 0:5g¼ minfAðxÞ;AðyÞ; 0:5g ð1Þ

In the following we show that Eq. (1) holds. It is clear that _i2K minfAiðxÞ;AiðyÞ; 0:5g 6minf_i2KAiðxÞ;_i2KAiðyÞ; 0:5g. If possible, let _i2K minfAiðxÞ, AiðyÞ; 0:5g 6¼ minf_i2KAiðxÞ;_i2KAiðyÞ; 0:5g. Thenthere exists r such that _i2K minfAiðxÞ, AiðyÞ, 0:5g < r< minf_i2KAiðxÞ;_i2KAiðyÞ; 0:5g. Since Ai � Aj or Aj � Ai

for all i; j 2 K, there exists k 2 K such that r < minfAkðxÞ, AkðyÞ; 0:5g. On the other hand, minfAiðxÞ;AiðyÞ; 0:5g < r for all i 2 K, a contradiction. Hence _i2K minfAiðxÞ;AiðyÞ; 0:5g ¼ minf_i2KAiðxÞ;_i2KAiðyÞ; 0:5g.

(2) For all x; a; y 2 S, we have

AðxayÞ ¼ ð[i2KAiÞðxayÞ ¼ _i2KAiðxayÞP _i2K minfAiðaÞ; 0:5gðsince Ai is an ð2;2 _qÞ-fuzzy interior ideal of S for all i 2 KÞ¼ minf_i2KAiðaÞ; 0:5g¼ minfAðaÞ; 0:5g

Therefore, A is an (2, 2 _q)-fuzzy interior ideal of S. h

3. A new proof of the necessity part of Theorem 4.12 in [1]

In the proof of the necessity part of Theorem 4.12 in [1], by the assumption, the authors concluded that‘‘8x; y 2 ½A�t; t 2 ð0; 1�, AðxyÞP minft; 0:5g’’ and that ‘‘8x; y 2 S; a 2 ½A�t; t 2 ð0; 1�, AðxayÞP minft; 0:5g’’.Note that these are incorrect as shown by the following example.

5800 Y. Yin, H. Li / Information Sciences 177 (2007) 5798–5800

Counterexample 3.1. Let S ¼ fa; b; cg be a semigroup with the following multiplication table:

Let A be a fuzzy subset of S such that

AðaÞ ¼ 0:6; AðbÞ ¼ 0:4; AðcÞ ¼ 0:3:

Then A is an (2, 2 _q)-fuzzy interior ideal of S. Now let t = 0.8. Then b; c 2 ½A�t, butAðbcÞ ¼ AðbÞ ¼ 0:4 < minft; 0:5g ¼ 0:5, AðbbcÞ ¼ AðbÞ ¼ 0:4 < minft; 0:5g ¼ 0:5.

Here, we present a new proof for the necessity part of Theorem 4.12 in [1].

A new proof. Let A be an (2, 2 _q)-fuzzy interior ideal of S and let x; y 2 ½A�t for t 2 ð0; 1�. Then xt 2 _qA andyt 2 _qA, that is, AðxÞP t or AðxÞ þ t > 1, and AðyÞP t or AðyÞ þ t > 1. Since A is an (2, 2 _q)-fuzzy interiorideal of S, we have AðxyÞP minfAðxÞ;AðyÞ; 0:5g.

Case 1 AðxÞP t and AðyÞP t. If t > 0.5, then AðxyÞP minfAðxÞ;AðyÞ; 0:5g ¼ 0:5, and thus ðxyÞtqA. If t 6 0.5,then AðxyÞP minfAðxÞ;AðyÞ; 0:5gP t, and so ðxyÞt 2 A.

Case 2 AðxÞP t and AðyÞ þ t > 1. If t > 0.5, then AðxyÞP minfAðxÞ;AðyÞ; 0:5g ¼ minfAðyÞ; 0:5g >minf1� t; 0:5g ¼ 1� t, i.e., AðxyÞ þ t > 1, and thus ðxyÞtqA. If t 6 0.5, then AðxyÞPminfAðxÞ;AðyÞ; 0:5gP minft; 1� t; 0:5g ¼ t, and so ðxyÞt 2 A.

Case 3 AðxÞ þ t > 1 and AðyÞP t. If t > 0.5, then AðxyÞP minfAðxÞ;AðyÞ; 0:5g ¼ minfAðxÞ; 0:5g >minf1� t; 0:5g ¼ 1� t, i.e., AðxyÞ þ t > 1, and thus ðxyÞtqA. If t 6 0.5, then AðxyÞPminfAðxÞ;AðyÞ; 0:5gP minf1� t; t; 0:5g ¼ t, and so ðxyÞt 2 A.

Case 4 AðxÞ þ t > 1 and AðyÞ þ t > 1. If t > 0.5, then AðxyÞP minfAðxÞ;AðyÞ; 0:5g > minf1� t; 0:5g ¼ 1� t,i.e., AðxyÞ þ t > 1, and thus ðxyÞtqA. If t 6 0.5, then AðxyÞP minfAðxÞ;AðyÞ; 0:5gP minf1� t; 0:5g ¼0:5 P t, and so ðxyÞt 2 A.Thus, in any case, we have ðxyÞt 2 _qA, and so xy 2 ½A�t. Hence ½A�t is a subsemigroup of S.Now, letx; y 2 S and a 2 ½A�t. Then at 2 _qA, that is, AðaÞP t or AðaÞ þ t > 1. Since A is an (2, 2 _q)-fuzzyinterior ideal of S, we have AðxayÞP minfAðaÞ; 0:5g.

Case 1 AðaÞP t. If t > 0.5, then AðxayÞP minfAðaÞ; 0:5g ¼ 0:5, and so AðxayÞ þ t > 0:5þ 0:5 ¼ 1. HenceðxayÞtqA. If t 6 0.5, then AðxayÞP minfAðaÞ; 0:5gP t, hence ðxayÞt 2 A.

Case 2 AðaÞ þ t > 1. If t > 0.5, then AðxayÞP minfAðaÞ; 0:5g > f1� t; 0:5g ¼ 1� t, i.e., AðxayÞ þ t > 1, andthus ðxayÞtqA. If t 6 0.5, then AðxayÞP minfAðaÞ; 0:5g > f1� t; 0:5g ¼ 0:5 P t, and so ðxayÞt 2 A.Thus, in both cases, we have ðxayÞt 2 _qA, and so xay 2 ½A�t.Therefore, ½A�t is an interior ideal of S for all t 2 ð0; 1�.

Acknowledgements

The authors are highly grateful to referees and Prof. Witold Pedrycz, Editor-in-Chief, for their valuablecomments and suggestions for improving the paper. This research was supported by National Natural ScienceFoundation of China (60474023) and Major State Basic Research Development Program of China(2002CB312200).

Reference

[1] Y.B. Jun, S.Z. Song, Generalized fuzzy interior ideals in semigroups, Inform. Sci. 176 (2006) 3079–3093.