OnMinimal Fuzzy Ideals of Semigroups

e present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. e existence of a fuzzy kernel in semigroup is explored. It has been shown that every fuzzy ideal of a semigroup contains everyminimal fuzzy le and every minimal fuzzy right ideal of semigroup. e fuzzy kernel is the class sum of minimal fuzzy le (right) ideals of a semigroup. Every fuzzy le ideal of a fuzzy kernel is also a fuzzy le ideal of a semigroup. It has been shown that the product of minimal fuzzy le ideal and minimal fuzzy right ideal of a semigroup forms a group. e representation of minimal fuzzy le (right) ideals and also the representation of intersection of minimal fuzzy le ideal and minimal fuzzy right ideal are shown. e fuzzy kernel of a semigroup is basically the class sum of all the minimal fuzzy le (right) ideals of a semigroup. Finally the sufficient condition of fuzzy kernel to be completely simple semigroup has been proved.


Introduction
Zadeh in 1965 introduced the fundamental concept of a fuzzy set in his paper [1] which provides a useful mathematical tool for describing the behavior of systems that are too complex or ill-de�ned to admit precise mathematical analysis by classical methods.e literature in fuzzy set theory and its practicability has been functioning quickly uptil now.e applications of these concepts can now be seen in a variety of disciplines like arti�cial intellegence, computer science, control engineering, expert systems, operation research, management science, and robotics.
Mordeson [2] has demonstrated the basic exploration of fuzzy semigroups.He also gave a theoratical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy �nite state machines, and fuzzy languages.e role of fuzzy theory in automata and daily language has extensively been discussed in [2].
Fuzzy sets are considered with respect to a nonempty set .e main idea is that each element  of  is assigned a membership grade  in [0, 1], with   0 corresponding to nonmembership, 0 <  < 1 to partial membership, and   1 to full membership.Mathematically, a fuzzy subset  of a set  is a function from  into the closed interval [0, 1].e fuzzy theory has provided generalization in many �elds of mathematics such as algebra, topology, di�erential equation, logic, and set theory.e fuzzy theory has been studied in the structure of groups and groupoids by Rosenfeld [3], where he has de�ned the fuzzy subgroupoid and the fuzzy le (right, two-sided) ideals of a groupoid .He has used the characteristic mapping of a subset  of a groupoid  to show that  is subgroupoid or le (right, two-sided) ideal of  if and only if   is the fuzzy subgroupoid or fuzzy le (right, two-sided) ideal of .Fuzzy semigroups were �rst considered by Kuroki [4] in which he studied the bi-ideals in semigroups.He also considered the fuzzy interior ideal of a semigroup, characterization of groups, union of groups, and semilattices of groups by means of fuzzy bi-ideals.N. Kehayopulu, X. Y. Xie, and M. Tsingelis have produced several papers on fuzzy ideals in semigroup.
A nonempty subset  of a semigroup  is a le (right) ideal of  if      and is ideal if  is both le and right ideal of .An ideal  of  is called minimal ideal of  if  does not properly contains any other ideal of .If the intersection  of all the ideals of a semigroup  is nonempty then we shall call  the kernel of .Huntington in [5] has shown the simpli�ed de�nition of a semigroup to be a group.We have used this concept in fuzzy semigroups and have shown that the product of minimal fuzzy le ideal and minimal fuzzy right ideal forms a group.Clifford in [6] had given the representation of minimal le (right) ideal of a semigroup.We have used this ideas to extend the fuzzy ideal theory of minimal fuzzy ideal in semigroups and have given a representation of fuzzy kernel in terms of minimal fuzzy le (right) ideals of a semigroup.
For a semigroup ,  will denote the collection of all fuzzy subsets of .Let  and  be two fuzzy subsets of a semigroup .e operation "∘" in  is de�ned by� such that  = , 0, otherwise. ( For simplicity, we will denote  as .

Sufficient Condition of a Semigroup to Be a Group
e very �rst de�nition of fuzzy point is given by �u and Liu [7].Let  be a nonempty set and   0, .A fuzzy point   of  means that   is a fuzzy subset of  de�ned by Each fuzzy point   of  is the fuzzy subset of a set  [7].Every fuzzy set  can be expressed as the union of all the fuzzy points in .It is to be noted that for any  and  in , , and  in 0,  the fuzzy points   and   have the inclusion relation that is   ⊆   if and only if  =  and   . In ∘    ∘   =   ∘   .
Now, we will show that if for any fuzzy point   of ,   ∘   =   then for each fuzzy point   of ,   ∘   =   .By the condition (ii) for fuzzy points   and   of  there exists a fuzzy point   of  such that   ∘   =   .Now by hypothesis, erefore (6) implies that   ∘   =   , and so   =   .It has been shown that the le cancellative law holds in .
Similarly, we can show that for any fuzzy points   and   of  there exists a unique fuzzy point   in , such that   ∘   =   .Similarly, we can show that  is a right cancellative.
Hence the cancellation law holds in a semigroup  and thus for each fuzzy point   of  we have,   ∘  =  a nd  ∘   = .Now there exists fuzzy points   and   of  such that   ∘   =   and   ∘   =   .Let   be any arbitrary fuzzy point of  then   is in  ∘   , that is there exists   in  such that   =   ∘   .us,   ∘   =   ∘    ∘   =   ∘   ∘    =   ∘   =   , which implies that   is the right identity in .Similarly we can show that   is the le identity in .Now since   is the le identity so   ∘   =   and   is the right identity so   ∘   =   .Hence   =   , which implies that  has an identity element   in .Now for   in , there exists   and   in  such that   ∘   =   and   ∘   =   .Now Hence there exists an inverse element for each fuzzy point of .e uniqueness of identity and inverse elements can be proved easily.
e following corollary can be easily asserted from the proof of eorem 1.

Corollary 2.
A fuzzy semigroup  is a fuzzy group if and only if   ∘  =  and  ∘   =  for all fuzzy points   of .Proof.Let  be the fuzzy le ideal of , then by eorem 12,  has a fuzzy kernel , where  is the class sum of all the minimal le ideals of .Now    is a fuzzy le ideal of , since               , and also     .us    is one of the minimal fuzzy le ideals of  or contains some minimal fuzzy le ideals of .But  is also contained in .Hence  contains at least one minimal fuzzy le ideal of .

Minimal Fuzzy Ideals of a Semigroup
Remark 16.Lemmas 10 and 11 and eorems 12, 13, and 15 are also applicable for fuzzy right ideal of a semigroup.
eorem 17.If a semigroup  contains at least one minimal fuzzy le ideal and one minimal fuzzy right ideal of , then the class sum of all the minimal fuzzy le ideals of  coincide with the class sum of all the minimal fuzzy right ideals and constitutes the fuzzy kernel of .
Proof.By eorem 12 the class sum of all the minimal fuzzy le ideals of  is a fuzzy kernel of .Similarly the class sum of all the minimal fuzzy right ideals of  is also the fuzzy kernel of .But fuzzy kernel of a semigroup  is unique since it is the intersection of all the fuzzy ideals of .
From now on we assume that   and   will denote the minimal fuzzy right ideal and the minimal fuzzy le ideal of a semigroup , respectively.A fuzzy idempotent point   of semigroup  is said to be another fuzzy idempotent Proof.Every fuzzy kernel of a semigroup is simple.Each fuzzy point   of the fuzzy kernel   of  belongs to exactly one minimal fuzzy le ideal   and to exactly one minimal fuzzy right ideal   of , otherwise   and   are not the minimal fuzzy le and minimal fuzzy right ideals of .So the fuzzy point   belongs to unique fuzzy group      .us   is the union of the disjoint fuzzy groups      .Each fuzzy idempotent point of   must be in one of these fuzzy groups, which can only have the fuzzy idempotent point   , namely the identity point.But by Lemma 23 the identity point is fuzzy primitive.e second condition for   to be completely simple is straightforward as each fuzzy point   of   is in one of the fuzzy groups      having fuzzy idempotent   such that             .Hence   is completely simple semigroup.
the following text for , , , , , , , and  in 0, , we denote   ,   ,   ,   ,   ,   , and   as the fuzzy points of different fuzzy subsets of .foranyfuzzypointsand   of  there exists   in , such that   ∘   =   , (ii) for any fuzzy points   and   of  there exists   in , such that   ∘   =   , then , ∘ is a fuzzy group.Proof.First of all we will show that the fuzzy points   and   existing in the hypothesis are uniquely determined.To show that   in condition (i) of the theorem is unique, let us assume on contrary that there exists two fuzzy points   and   such that,   ∘   =   and   ∘   =   .us,∘=   ∘   .(4)econdition (ii) in the theorem implies that for the fuzzy points   and   of  there exists a fuzzy point   of  such that   ∘   =   .Hence (4) implies that,   ∘   ∘    =   ∘   , [2]nonempty intersection of fuzzy ideals of semigroup  is called the fuzzy kernel of a semigroup .efuzzyideal  of  is called minimal fuzzy ideal if  contains no proper fuzzy ideal of .A semigroup  is called a fuzzy simple if it does not contain any proper fuzzy ideal of .efollowinglemmasare in our previous knowledge[2].For any nonempty subsets  and  of semigroup , we have    if and only if      .Let  be a nonempty subset of a semigroup , then  is an ideal of  if and only if   is a fuzzy ideal of .Let  and  be fuzzy ideals of a semigroup , then    and    are also fuzzy ideals of .Let  and  be fuzzy ideals of a semigroup , then    and    are also fuzzy ideals of .  is a fuzzy ideal of .Suppose that   is not minimal fuzzy ideal of , then there exists some fuzzy ideal   of  such that,      .Hence by Lemma 3,   , where  is an ideal of .is is a contradiction to the fact that  is minimal ideal of .us   is the minimal fuzzy ideal of .Conversely, let   be the minimal fuzzy ideal of , then by Lemma 4,  is the ideal of .Suppose that  is not minimal ideal of , then there exists some ideal  of  such that   .Now by Lemma 3,      , where   is the fuzzy ideal of .is is a contradiction to the fact that   is the minimal fuzzy ideal of .us  is the minimal ideal of .Proof.Since  is a fuzzy ideal of , so  is a fuzzy subsemigroup of , since         .To show  is simple, let  be any fuzzy ideal of , then      is fuzzy ideal of , since                      and     .Also     , but by eorem 8,  is minimal fuzzy ideal of  because every fuzzy kernel of , if exists, is a minimal fuzzy ideal of .Hence       .Also           , which implies that   .us   , implies that  is simple subsemigroup of .Since for a minimal fuzzy le ideal  of ,                    , which implies that     is a fuzzy le ideal of .Suppose  is the fuzzy le ideal of     and let ℎ  {  ∈          for  ∈ [0, 1].Let for  in [0, 1],   be a fuzzy point of  then for      in ℎ,      is in  and so          .Now       ℎ, which implies that   ℎ  ℎ.Hence ℎ is fuzzy le ideal of  contained in minimal fuzzy le ideal  of  and so ℎ  .us for all   in ,       , which implies that      .Hence      , and so     is a minimal fuzzy le ideal of .Proof.Let  be any minimal fuzzy le ideal of , then    is fuzzy le ideal of  since,               .Also      as  is a fuzzy le ideal of .But  is minimal so       .Hence every minimal fuzzy le ideal  of  is contained in every fuzzy ideal of .   1 ∪ 2 ∪ 3 ∪⋯   1 ∪ 2 ∪  3 ∪⋯   1 ∪ 2 ∪ 3 ∪⋯  , which implies that  is the fuzzy le ideal of .Now it has to be shown that  is minimal.Let, for ,  in [0, 1],   and   be arbitrary fuzzy points of  and , respectively.By de�nition of , the fuzzy point   is in some   for some  ∈ , such that            .Now since   is minimal fuzzy le ideal of  so by Lemma 10,      is also minimal fuzzy le ideal of  and hence contained in .us,        and so     .Hence  is fuzzy right ideal of  and so fuzzy ideal of .Now by Lemma 11, every fuzzy ideal contains every minimal fuzzy le ideal of  and so  being union of all minimal fuzzy le ideals of  is contained in every fuzzy ideal.Hence  is the fuzzy kernel of .Proof.Let  be the fuzzy kernel of , then by eorem 12,  is the class sum of all the minimal fuzzy le ideals of .Let  be a fuzzy le ideal of , then each fuzzy point   for  in [0, 1], of  belongs to some minimal fuzzy le ideal   ,  ∈  of .Now     is a fuzzy le ideal of , since                    .Also             , where   is minimal fuzzy le ideal so        .us        implies that       , where     is minimal fuzzy le ideal of  and so       .Hence,  is the fuzzy le ideal of .Let a semigroup  contain at least one minimal fuzzy le ideal of  then every fuzzy le ideal of  contains at least one minimal fuzzy le ideal of .
eorem 7. A nonempty subset  of a semigroup  is minimal ideal if and only if   is minimal fuzzy ideal of .Proof.Let  be a minimal ideal of , then by Lemma 4, eorem 8.If a semigroup  contains a minimal fuzzy ideal  of  then  is the fuzzy kernel of .Proof.Let  be any fuzzy ideal of  then for a minimal fuzzy ideal  of ,       .us    is non-empty.Since by Lemma 6,    is a fuzzy ideal of  and     , which implies that     .But then       , so  contains in every fuzzy ideal of  and hence is a fuzzy kernel of .eorem9.If a semigroup has fuzzy kernel  then  is a simple subsemigroup of .Lemma 10.If  is a minimal le ideal of a semigroup  and   be any fuzzy point of  for  in [0, 1] then     is also a minimal fuzzy le ideal of .Proof.eorem 12.If a semigroup  contains at least one minimal fuzzy le ideal then it has a fuzzy kernel of , where fuzzy kernel is the class sum of all the minimal fuzzy le ideals of .Proof.Let  be the union of all the minimal fuzzy le ideals of .Since  contains at least one minimal fuzzy le ideal of , so  is non-empty.Let  1 ,  2 ,  3 , … be all the minimal fuzzy le ideals of  then, Remark 14.Every minimal fuzzy le ideal of  is a minimal fuzzy le ideal of fuzzy kernel of  and vice versa.eorem 15.
point   if               .e fuzzy idempotent   is called fuzzy primitive if there is no fuzzy idempotent under   .A simple fuzzy semigroup  is said to be completely simple if every fuzzy idempotent point of  is fuzzy primitive and for each fuzzy point   of  there exists fuzzy idempotents   and   such that               .Let   be in   and   be in      , then there exists a fuzzy point   in      such that         .Proof.Note that      is a fuzzy right ideal of  since,                  and also             .But   is the minimal fuzzy right ideal of  so         .Hence               .Let   be in   and   be in      , then there exists a fuzzy point   in      such that         .Proof.Since     is a fuzzy le ideal of  since,                     and also               .But   is the minimal fuzzy le ideal of  so         .Hence               .eorem 20.     is a group.Proof.Since                              , which implies that     is a semigroup.Let   and   be any two fuzzy points of      then   and   are in both   and   , since        and        .us by Lemmas 18 and 19 there exists   and   in  such that,         and         .Hence by eorem 1,      is a group.Let   be the identity element of the group      , , then       ,        and   ∩           .Proof.Since identity element is in   since        .Now                  , which implies that     is a fuzzy right ideal of  and is contained in minimal fuzzy right ideal   of  and hence       .Similarly we can show that        .Now                                   , also                      .eorem 22.For minimal fuzzy right ideal   and minimal fuzzy le ideal   of a semigroup ,        ∩   .Proof.We only need to show that   ∩         .Let   be arbitrary fuzzy point of   ∩   and let   be identity element of      then by Lemma 18 there exists fuzzy point   of      such that         .Let   be the inverse fuzzy element of   in      then,                                   .Now            , so by Lemma 21,         .Hence      and so   is in      .Lemma 23. e identity fuzzy point   of the fuzzy group      is a fuzzy primitive.Proof.Let   be a fuzzy idempotent under   , then               .Further, Lemma 21 implies that   is in   ∩   .By eorem 22   is in the fuzzy group      , which can contain only one fuzzy idempotent point, namely the identity element   , so      .eorem 24.Let a semigroup  have at least one minimal fuzzy le ideal and at least one minimal fuzzy right ideal of  then the fuzzy kernel   of  is completely simple semigroup.