Fuzzy subgroups and anti fuzzy subgroups

Fuzzy Sets and Systems 35 (1990) 121-124 North-Holland

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SHORT COMMUNICATION F U Z Z Y S U B G R O U P S A N D ANTI F U Z Z Y S U B G R O U P S R. BISWAS Department of Mathematics, B.B.E. College, Agartala 799004, Tripura, India Received October 1988 Revised November 1988

Abstract: We introduce the concept of anti fuzzy subgroups of groups, lower level subset of a fuzzy subset, lower level subgroups and prove some results on these. We show that a fuzzy subset of a group is a fuzzy subgroup iff the complement of this fuzzy subset is anti fuzzy subgroup. We fuzzify lower level subsets and prove that if a fuzzy set is an anti fuzzy subgroup then the fuzzifications of its lower level subsets are also anti fuzzy subgroups.

Keywords: Fuzzy set; fuzzy subgroup; anti fuzzy subgroup.

1. Introduction The concept of fuzzy sets was initiated by Zadeh [1]. Since then it has become a vigorous area of research in engineering, medical science, social science, physics, statistics, graph theory, etc. Rosenfeld [2] gave the idea of fuzzy subgroups. P. Das [3] studied level subgroups. In the present paper we study the 'lower level subset of a fuzzy set', 'anti fuzzy subgroups', 'lower level subgroups' and prove some results on these.

2. helimima-ies Here we give a brief review of some preliminaries. Definition 2.1. If S is any set, a mapping # : S--* [0, 1] is called a fuzzy subset of S. Definition 2.2. Let # be a fuzzy subset of S. Then for t ~ [0, 1], the level subset of the fuzzy set # is the set #t = {x e S : # ( x ) > ~ t } .

Definition 2.3. Let G be a group. A fuzzy subset # of G is called a fuzzy 0165-0114190/$3.50 ( ~ 1990, Elsevier Science Publishers B.V. (North-Holland)

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subgroup of G if for x, y • G: (i) ~t(xy) >>-min{/z(x),/~(y)},

(ii) /z(x -1) t> #(x). Note that here/t(x) ~>-min{p(x),/z(y)}.

Definition 2.4. If gt is a fuzzy subset of S, then the complement of ~, denoted by gt¢, is the fuzzy set of S given by lt¢(x) = l - ~t(x)

Vx e S.

3. Anti fuzzy subgroups Definition 3.1. Let G be a group. A fuzzy subset/t of G is called an anti fuzzy subgroup of G if for x, y • G: (i) gt(xy) <<-max{p(x), ~t(y)}, (ii) ~t(x-1) <~/~(x). The following proposition is clear from above definition. Proposition 3.1. If tz is an anti f u z z y subgroup of a group G, then Vx • G:

(i) /z(e) ~>-min{/~(x),/z(y)} or,

1 - gC(xy) >! min{(1 - #C(x)), (1 - #C(y))} or,

p~(xy) ~< 1 - man{(1 -/z~(x)), (1 - #~(y))} or,

#C(xy) <<,max{p~(x), ~C(y)}.

Also, or, 1 -

>! 1 -

or, g°(x-') -< g°(x). Thus/~¢ is an anti fuzzy subgroup. The converse also can be proved similarly.

Fuzzy subgroups and anti fuzzy subgroups

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Proposition 3.3. It/s an anti f u z z y subgroup o f a group G iff Vx, y • G,

#(xy -l) <~max(it(x), It(y)}.

4. Lower level subsets Definition 4.1. Let It be a fuzzy set of S. For t • [0, 1], the lower level subset of It is the set ~t = {X • S : it(x) <~t}.

C l e a r l y , / ~ = S and #t U/~t = S. If tl < t2, then/at, ~_/~t2Proposition 4.1. Let It be an anti f u z z y subgroup o f a group G. Then for t • [0, 1] such that t >t It(e), ftt is a subgroup of G. ProoL Vx, y •/~t, we have

It(xy -1) <--max{it(x), It(y)} = t, which concludes the proof. Proposition 4.2. Let G be a group and It be a f u z z y subset of G such that (tt is subgroup of G Vt • [0, 1] with t >t It(e). Then !z is an anti f u z z y subgroup o f G.

a

Proof. Let x, y • G and It(x) = tl, It(y) = t2. Suppose tl ~
It(xy -l) ~ It(e), are called lower level subgroups of It. The following proposition is straightforward: Proposition 4.3. Let It be an anti f u z z y subgroup of G. Two lower level subgroups #t. and (tt2 (with tl < t2) o f lz are equal iff there is no x • G such that tl < It(x) < t2.

5. Fuzzilication of a lower level subset Definition 5.1. Let # be a fuzzy set of a set S. Then the lower level subset is #, = {x • S : i t ( x ) ~
where t • [0, 1].

Now fuzzification of/at is the fuzzy set A~, defined by

Ar,,(x ) = It(x) = 0

ifx •/~t otherwise.

Clearly, A~, _~ It and (At,,) t = [~,.

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Proposition 5.1. I f I~ is an anti f u z z y subgroup of G, then A~,, is also an anti f u z z y subgroup o f G where t ~ [0, 1], and t >! bt(e). Proof. Clearly/~t is a subgroup of G. Let x, y ~ G and suppose x, y e/~,. Then mf,,(xy -1 ) = / ~ ( x y -1)

~
Ar~,(xy -1) = 0 <~max{A~,,(x ), a~,(y)} Hence xy -1 ~ A~,. Finally, suppose x, y ~/zt- In this case we can also prove that xy -1 ~ A~,. Hence the proposition is proved.

References [1] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [2] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [3] P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269.