GAZ˙IOSMANPAS¸A UNIVERSITY
GRADUATE SCHOOL of NATURAL and APPLIED SCIENCES
SOFT UNION SEMIGROUPS Aslıhan Sezgin SEZER
Ph. D. Thesis
Department of Mathematics Assoc. Prof. Dr. Naim C¸ A ˘GMAN
2012
GAZ˙IOSMANPAS¸A UNIVERSITY
GRADUATE SCHOOL of NATURAL and APPLIED SCIENCES DEPARTMENT OF MATHEMATICS
Ph. D. THESIS
SOFT UNION SEMIGROUPS
Aslıhan Sezgin SEZER
TOKAT 2012
I hereby declare that I conform the required standard thesis format, academic rules and ethical conduct. I also declare that, as required by these rules and conduct, all materials benefited in this thesis consist of the mentioned recourses in the reference list. I verify that no innovation and results in this thesis has been obtained anywhere else, no data used in this thesis is falsified and no parts of this thesis is presented as a thesis in this or any other university.
Ph. D. Thesis
SOFT UNION SEMIGROUPS Aslıhan Sezgin SEZER Gaziosmanpa¸sa University
Graduate School of Natural and Applied Sciences Department of Mathematics
Supervisor : Assoc. Prof. Dr. Naim C¸ A ˘GMAN
Soft set theory was initiated by Molodtsov in 1999 as a completely generic mathematical tool for modeling uncertainties. This theory has been applied to many fields such as information systems, decision making problems, optimization theory, etc. In this study, applying soft set theory we construct soft union semigroups and study their properties. First, the literature and basic concepts regarding soft sets are given. Then, soft union product, soft anti characteristic function and soft union ideals of semigroups are introduced and the interrelations of them are presented. Moreover, certain kind of semigroups are characterized in terms of soft union ideals of semigroups. Finally, soft normal semigroups are defined and some characterizations of semigroups with soft normality are given.
2012, 146 pages
Keywords: Soft sets, Soft union left ideals, Soft union bi-ideals, Soft union interior ideals, Soft union quasi-ideals, Regular semigroups, Semisimple semigroups, Duo semigroups.
OZET
Doktora Tezi
ESNEK B˙IRLES¸ ˙IMSEL YARIGRUPLAR Aslıhan Sezgin SEZER
Gaziosmanpasa ¨Universitesi Fen Bilimleri Enstit¨us¨u Matematik Anabilim Dalı
Danı¸sman : Do¸c. Dr. Naim C¸ A ˘GMAN
Esnek k¨ume teorisi, Molodtsov tarafından 1999 yılında belirsizlikleri modellemek i¸cin genel bir matematiksel model olarak sunulmu¸stur. Bu teori, bilgi sistemleri, karar verme problemleri, optimizasyon teorisi gibi belirsizlik i¸ceren bir ¸cok alana uygulandı. Bu tez ¸calısmasında, esnek k¨ume teorisi esnek birle¸simsel yarıgruplara uygulanarak, esnek birle¸simsel yarıgruplar in¸sa edilip ¨ozellikleri ¸calı¸sıldı. ˙Ilk olarak, esnek k¨umeler ile ilgili litarat¨ur taraması ve temel bilgiler verildi. Daha sonra, esnek birle¸simsel ¸carpım, esnek anti karakteristik fonksiyonlar ve yarıgrupların esnek birle¸simsel idealleri tanımlandı ve birbirleriyle olan ili¸skileri sunuldu. Ayrıca, bazı yarıgrup ¸ce¸sitleri, esnek birle¸simsel yarıgrupların idealleri kullanılarak karakterize edildi. Son olarak, esnek normal yarıgruplar tanımlanarak, esnek normallik ile ilgili bazı karakterizasyonlar verildi.
2012, 146 sayfa
Anahtar Kelimeler: Esnek k¨ume, Esnek birle¸simsel ideal, Esnek birle¸simsel iki-ideal, Esnek birle¸simsel orta ideal, Esnek birle¸simsel idealimsi, Reg¨uler yarıgrup, Yarıbasit yarıgrup, ˙Ikili yarıgrup.
ABSTRACT . . . i
¨ OZET . . . ii
PREFACE . . . v
SYMBOL and ABBREVIATION INDEX . . . vi
1. INTRODUCTION . . . 1
1.1 Literature Summary . . . 2
1.2 Material and Method . . . 4
2. BASIC CONCEPTS . . . 6
3. CONSTRUCTION OF SOFT UNION SEMIGROUPS . . . 9
3.1 Soft Union Product and Soft Anti Characteristic Function . . . 9
3.2 Soft Union Semigroups . . . 12
4. IDEALS OF SOFT UNION SEMIGROUPS . . . 18
4.1 Soft Union Ideals of Semigroups . . . 18
4.2 Soft Union Bi-ideals of Semigroups . . . 26
4.3 Soft Union Interior Ideals of Semigroups . . . 30
4.4 Soft Union Quasi-ideals of Semigroups . . . 41
4.5 Soft Union Generalized Bi-ideals of Semigroups . . . 47
5. CERTAIN CLASSES OF SEMIGROUPS CHARACTERIZED BY SOFT UNION IDEALS . . . 51
5.1 Regular Semigroups . . . 51
5.2 Intra-regular Semigroups . . . 70
5.3 Completely Regular Semigroups . . . 86
5.4 Quasi-regular Semigroups . . . 90
5.5 Weakly Regular Semigroups . . . 93
5.6 Semisimple Semigroups . . . 96
5.7 Regular Duo Semigroups . . . 99
5.8 Right and Left Zero Semigroups . . . 105
5.9 Semilattices of Left or Right Simple Semigroups . . . 108
5.10 Semilattice of Left or Right Groups . . . 112
5.11 Semilattice of Groups . . . 125
6. SOFT NORMAL SEMIGROUPS . . . 135
7. CONCLUSION . . . 140
REFERENCES . . . 141 iii
I wish to express my special and profound gratitude to my supervisor Assoc. Prof. Dr. Naim C¸ A ˘GMAN, who has been a source of motivation and inspiration for me.
I am grateful to Assoc. Prof. Dr. Akın Osman ATAG ¨UN with whom I have had fruitful discussions for improving this study.
I would like to thank to Prof. Dr. Oktay MUHTARO ˘GLU for his invaluable comments, suggestions and guidance.
I am also thankful to my beloved husband, Mustafa Sabri SEZER, whose moral support and constant love gave me the courage to complete this thesis.
This thesis was financially supported by The Sciencific and Technological Research Counsil of Turkey (T ¨UB˙ITAK) via the grant of 2211 Domestic Doctoral Program. I would like to thank to T ¨UBITAK for its financial support.
Aslıhan Sezgin SEZER November 2012
S Semigroup
U Initial universe set E Set of parameters P (U ) Power set of U
S(U ) The set of all soft sets over U fA∪fe B Union of fA and fB
fA∩fe B Intersection of fA and fB
Ψ?(f
A) Anti image of fA under Ψ
Ψ−1(fB) Soft pre-image or soft inverse image of fB under Ψ
L(fA; α) Lower α-inclusion of fA
fS∗ gS Soft union product of fS and gS
e
θ Null soft union semigroup and ideals of semigroups SXc Soft characteristic function of the complement X
L[a] Principal left ideal of S generated by a R[a] Principal right ideal of S generated by a
J [a] Principal ideal of S generated by a B[a] Principal bi-ideal of S generated by a Q[a] Principal quasi-ideal of S generated by a
I[a] Principal interior ideal of S generated by a
Modern set theory formulated by George Cantor is fundamental for the whole of Mathematics. One issue associated with the notion of a set is the concept of vagueness. Mathematics requires that all mathematical notions including set must be exact. This vagueness or the representation of imperfect knowledge has been a problem for a long time for philosophers, logicians and mathematicians. However, recently it became a crucial issue for computer scientists, particularly in the area of artificial intelligence. To solve complicated problems involving various uncertainties in economics, engineering and environment, we cannot successfully use classical methods. To handle situations like this, many tools have been suggested. They include, theory of probability, fuzzy set theory (Zadeh, 1965), interval mathematics and rough set theory (Pawlak, 1982), which we can consider as mathematical tools for dealing with imperfect knowledge. All these tools require the pre-specification of some parameter to start with, e.g. probability density function in probability theory, membership function in fuzzy set theory and an equivalence relation in rough set theory. Moreover, probability theory is applicable only for a stochastically stable system. Interval mathematics is not sufficiently adaptable for problems with different uncertainties. Setting the membership function value is always been a problem in fuzzy set theory. Furthermore, all these techniques lack in the parametrization of the tools and hence they could not be applied successfully in tackling problems especially in areas like economic, environmental and social problem domains.
Soft theory was initiated by the Russian researcher Molodtsov (1999). Molodtsov proposed the soft set as a completely generic mathematical tool for modeling uncertainties. There is no limited condition to the description of objects; so researchers can choose the form of parameters they need, which greatly simplifies the decision-making process and make the process more efficient in the absence of partial information.
Moreover, soft set theory does not require the specification of a parameter, instead it accommodates approximate descriptions of an object as its starting point. This
makes the theory a natural mathematical formalism for approximate reasoning. We can use any parametrization we prefer: with the help of words, sentences, real numbers, functions, mappings, and so on. This means that the problem of setting the membership function or any similar problem does not arise in soft set theory. Soft set theory is standing in a unique way in the sense that it is free from the above difficulties and has a wider scope for many applications in a multidimensional way, some of which are reported by Molodtsov (1999, 2004) in his studies. He successfully applied soft set theory in areas such as the smoothness of functions, game theory, operation research, Riemann integration and so on.
A soft set can be considered as an approximate description of an object precisely consisting of two parts, namely predicate and approximate value set. Exact solutions to the mathematical models are needed in classical mathematics. If the model is so complicated that we cannot set an exact solution, we can derive an approximate solution and there are many methods for this. On the other hand, in soft set theory as the initial description of object itself is of an approximate nature, we need not have to introduce the concept of an exact solution.
1.1 Literature Summary
Soft set proposed by the Russian researcher Molodtsov (1999) as a mathematical tool for modeling uncertainties has received much attention in mathematics both in theory and application. Maji et al. (2002) presented an application of soft sets in a decision making problem and published a detailed theoretical study on soft sets. Chen et al. (2003) and Xiao et al. (2003) presented research on synthetically evaluating a method for business competitive capacity based on soft sets. Chen et al. (2005) and Kong et al. (2008) introduced a new definition of soft set parametrization reduction. Xiao et al. (2005) and Pei and Miao (2005) discussed the relationship between soft sets and information systems. Mushrif et al. (2006)
presented a new algorithm for classification of the natural textures. The proposed classification algorithm is based on the notions of soft set theory.
However, works on soft set theory has been progressing rapidly since Maji et al. (2003) introduced several operations of soft sets. Since then, Pei and Miao (2005), Ali et al. (2009) and Sezgin and Atag¨un (2011a) introduced and studied on several soft set operations as well. The algebraic structure of soft set theories has been studied by many authors, too. Akta¸s and C¸ a˘gman (2007) gave the definition of soft groups. They also compared soft sets to the related concepts of fuzzy sets and rough sets. Jun (2008) applied Molodtsov’s notion of soft sets to the theory of BCK/BCI-algebras and introduced the notion of soft BCK/BCI-algebras and soft subalgebras and then investigated their basic properties. Jun and Park (2008) dealt with the algebraic structure of BCK/BCI-algebras by applying soft set theory. They introduced the notion of soft ideals and idealistic soft BCK/BCI-algebras and gave several examples. Jun et al. (2009) introduced the notion of soft p-ideals and p-idealistic soft BCI-algebras and investigated their basic properties. Using soft set, they gave characterization of (fuzzy) p-ideals in BCI-algebras. Feng et al. (2008) introduced soft semirings, soft subsemirings, soft ideals, idealistic soft semirings, and soft semiring homomorphisms. Acar et al. (2010) introduced initial concepts of soft rings and defined soft subrings, soft ideals, idealistic soft rings, and soft ring homomorphisms, together with their related properties. Sun et al. (2008) introduced a basic version of soft module theory, which extends the notion of a module by including some algebraic structures in soft sets. Sezgin et al. (2011) studied soft near-rings and idealistic soft near-rings. The soft substructures of rings, fields, and modules were first introduced by Atag¨un and Sezgin (2011). There are also some other significant papers on soft set theory such as Babitha et al. (2010), Babitha et al. (2012), Feng et al. (2010), Gong et al. (2010), Majumdar and Samanta (2008), Park et al. (2008), Park et al. (2012), Sezgin and Atag¨un (2011b), Sezgin et al. (2012a), Sezgin et al. (2012b). To develop soft set theory, the operations of the soft sets were redefined and a uni-int decision making method was constructed by using these new operations by C¸ a˘gman and Engino˘glu (2010).
1.2 Material and Method
The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of ”memoryless” systems: time-dependent systems that start from scratch at each iteration. In applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata. In probability theory, semigroups are associated with Markov processes.
In this study, we bring a new approach to semigroup theory via the soft sets. By using the redefined soft set definitions of C¸ a˘gman and Engino˘glu (2010) and Sezgin and Atag¨un (2011a) and applying soft set theory to semigroups, we construct soft union semigroups and soft union ideals of semigroups and study their properties. This thesis consists of seven chapters. The literature summary, material and method and basic concepts have been given in the first and second chapters. In the third chapter, soft union product, soft anti characteristic function are defined and studied. In the fourth chapter, soft union semigroups, soft union left ideals, bi-ideals, interior ideals, quasi-ideals and generalized bi-ideals are defined and their properties with the interrelations of them are obtained. In the fifth chapter, certain kind of semigroups such as regular, intra-regular, completely regular, weakly regular, quasi-regular semigroups, semisimple semigroups, duo semigroups, right and left zero semigroups, right and left simple semigroups, semilattice of right and left simple semigroups, semilattice of right and left groups and semilattice of groups are characterized in terms of soft union ideals of semigroups. While studying especially on this chapter, we often use the books of Clifford and Preston (1961), Howie (1995) and Petrich (1973) as references for semigroups. In the last two chapters, soft normal semigroup
is defined, some characterizations of semigroups with soft normality is given and the conclusion is presented.
In this section, we recall some basic notions relevant to semigroups and soft sets. In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element. It also (originally) generalized a group (a monoid with all inverses) to a type where every element did not have to have an inverse, thus the name semigroup. Note that throughout this study, S denotes a semigroup.
A nonempty subset A of S is called a subsemigroup of S if AA ⊆ A, a right ideal of S if AS ⊆ A, a left ideal of S if SA ⊆ A. By two-sided ideal or simply ideal, we mean a subset of S, which is both a left and right ideal of S. A subsemigroup X of S is called a bi-ideal of S if XSX ⊆ X, an interior ideal of S if SXS ⊆ X, a quasi-ideal of S if XS ∩ SX ⊆ X. We denote by L[a], R[a], J [a], B[a], Q[a], I[a], the principal left, right, two-sided, bi-ideal, quasi-ideal, interior ideal of S generated by a ∈ S, that is, L[a] = {a} ∪ Sa, R[a] = {a} ∪ aS, J [a] = {a} ∪ Sa ∪ aS ∪ SaS, B[a] = {a} ∪ {a2} ∪ aSa, Q[a] = {a} ∪ (aS ∩ Sa), I[a] = {a} ∪ {a2} ∪ SaS, respectively.
A subset P of a semigroup S is called semiprime if for all a ∈ S, a2 ∈ P implies
that a ∈ P . A semilattice S is a structure S = (S, .), where “.” is an infix binary operation, called semilattice operation, such that “.” is associative, commutative and idempotent. A semigroup S is called regular if for every element a of S there exists an element x in S such that a = axa or equivalently a ∈ aSa.
For all undefined concepts and notions about semigroups, we refer to Clifford and Preston (1961), Howie (1995) and Petrich (1973).
From now on, U refers to an initial universe, E is a set of parameters, P (U ) is the power set of U and A, B, C ⊆ E.
Definition 2.0.1. A soft set fA over U is a set defined by
fA : E → P (U ) such thatfA(x) = ∅ if x /∈ A.
Here, fA is also called approximate function. A soft set over U can be represented
by the set of ordered pairs
fA= {(x, fA(x)) : x ∈ E, fA(x) ∈ P (U )}
(Molodtsov, 1999).
It is clear to see that a soft set is a parametrized family of subsets of the set U . It is worth noting that the sets fA(x) may be arbitrary. Some of them may be empty,
some may have nonempty intersection. If we define more than one soft set in a subset A of the set of parameters E, then the soft sets will be denoted by fA, gA, hA
etc. If we define more than one soft set in some subsets A, B, C etc. of parameters E, then the soft sets will be denoted by fA, fB, fC etc., respectively.
We refer to C¸ a˘gman and Engino˘glu (2010), Maji et al. (2003), Molodsov (1999) for further details. From now on, the set of all soft sets over U will be denoted by S(U ). Definition 2.0.2. Let fA, fB ∈ S(U ). Then, fA is called a soft subset of fB and
denoted by fA⊆f˜ B, if fA(x) ⊆ fB(x) for all x ∈ E (C¸ a˘gman and Engino˘glu, 2010).
Definition 2.0.3. Let fA, fB ∈ S(U ). Then, union of fA and fB, denoted by
fA∪fe B, is defined as fA∪fe B = fAe∪B, where fAe∪B(x) = fA(x) ∪ fB(x) for all x ∈ E
and intersection of fA and fB, denoted by fA∩fe B, is defined as fA∩fe B = fAe∩B,
where fAe∩B(x) = fA(x) ∩ fB(x) for all x ∈ E (C¸ a˘gman and Engino˘glu, 2010).
Definition 2.0.4. Let fA, fB ∈ S(U ). Then, ∨-product of fA and fB, denoted by
fA∧ fB, is defined as fA∨ fB = fA∨B, where fA∨B(x, y) = fA(x) ∪ fB(y) for all
Definition 2.0.5. Let fAand fB be soft sets over the common universe U and Ψ be
a function from A to B. Then, soft anti image of fA under Ψ, denoted by Ψ?(fA),
is a soft set over U defined by
(Ψ?(fA))(b) =
T{fA(a) | a ∈ A and Ψ(a) = b}, if Ψ−1(b) 6= ∅,
∅, otherwise
for all b ∈ B. And soft pre-image or soft inverse image of fB under Ψ, denoted by
Ψ−1(fB), is a soft set over U by (Ψ−1(fB))(a) = fB(Ψ(a)) for all a ∈ A (C¸ a˘gman et
al., 2011).
Definition 2.0.6. Let fA be a soft set over U and α ⊆ U . Then, lower α-inclusion
of fA, denoted by L(fA; α), is defined as
L(fA; α) = {x ∈ A | fA(x) ⊆ α}
In this chapter, we first define soft union product and soft anti characteristic function in order to construct soft union semigroups and soft union ideals of semigroups. Then, we introduce soft union semigroup and study its basic properties.
3.1 Soft Union Product and Soft Anti Characteristic Function
In this section, we define soft union product and soft anti characteristic function and study their properties.
Definition 3.1.7. Let fS and gS be soft sets over the common universe U . Then,
soft union product fS∗ gS is defined by
(fS∗ gS)(x) =
T
x=yz{fS(y) ∪ gS(z)}, if ∃y, z ∈ S such that x = yz,
U, otherwise
for all x ∈ S. Note that soft union product is abbreviated by soft uni-product in what follows.
Example 3.1.8. Consider the semigroup S = {a, b, c, d} defined by the following table: . a b c d a a a a a b a a a a c a a b a d a a b b
Let U = D2 = {< x, y >: x2 = y2 = e, xy = yx} = {e, x, y, yx} be the universal
{e, x}, fS(c) = {y, yx}, fS(d) = {e, x, y} and gS(a) = {x, y}, gS(b) = {e, yx}, gS(c) =
{yx}, gS(d) = {e, y}. Since b = cc = dc = dd, then
(fS ∗ gS)(b) = {fS(c) ∪ gS(c)} ∩ {fS(d) ∪ gS(c)} ∩ {fS(d) ∪ gS(d)} = {y}
Similarly, (fS∗ gS)(a) = ∅, (fS∗ gS)(c) = (fS∗ gS)(d) = U .
Theorem 3.1.9. Let fS, gS, hS ∈ S(U ). Then,
1) (fS ∗ gS) ∗ hS = fS∗ (gS∗ hS).
2) fS∗ (gS∪he S) = (fS∗ gS)∪(fe S∗ hS) and (fS∪ge S) ∗ hS = (fS ∗ hS)∪(ge S∗ hS). 3) fS∗ (gS∩he S) = (fS∗ gS)∩(fe S∗ hS) and (fS∩ge S) ∗ hS = (fS ∗ hS)∩(ge S∗ hS). 4) If fS⊆g˜ S, then fS∗ hS⊆g˜ S∗ hS and hS∗ fS⊆h˜ S∗ gS.
5) If tS, lS ∈ S(U ) such that tS⊆f˜ S and lS⊆g˜ S, then tS∗ lS⊆f˜ S∗ gS.
Proof . (1) follows from Definition 3.1.7 and Example 3.1.8.
(2) Let a ∈ S. If a is not expressible as a = xy, then (fS ∗ (gS∪he S))(a) = U . Similarly,
((fS∗ gS)∪(fe S∗ hS))(a) = (fS∗ gS)(a) ∪ (fS∗ hS)(a) = U ∪ U = U
Now, let there exist x, y ∈ S such that a = xy. Then,
(fS∗ (gS∪he S))(a) = \ a=xy (fS(x) ∪ (gS∪he S)(y)) = \ a=xy (fS(x) ∪ (gS(y) ∪ hS(y)) = \ a=xy [(fS(x) ∪ gS(y)) ∪ (fS(x) ∪ hS(y))] = [ \ a=xy (fS(x) ∪ gS(y))] ∪ [ \ a=xy (fS(x) ∪ hS(y))] = (fS∗ gS)(a) ∪ (fS∗ hS)(a) = [(fS∗ gS)e∪(fS∗ hS)](a)
Thus, (fS∪ge S) ∗ hS = (fS∗ hS)e∪(gS ∗ hS) and (3) can be proved similarly.
4) Let x ∈ S. If x is not expressible as x = yz, then (fS∗ hS)(x) = (gS∗ hS)(x) = U .
Otherwise, (fS∗ hS)(x) = \ x=yz (fS(y) ∪ hS(z)) ⊆ \ x=yz
(gS(y) ∪ hS(z)) (since fS(y) ⊆ gS(y))
= (gS ∗ hS)(x)
Similarly, one can show that hS∗ fS⊆h˜ S∗ gS. (v) can be proved similar to (4).
Note that, fS∗ gS is not equal to gS∗ fS, generally.
Definition 3.1.10. Let X be a subset of S. We denote by SXc the soft characteristic
function of the complement X and define as
SXc(x) = ∅, if x ∈ X, U, if x ∈ S \ X
Theorem 3.1.11. Let X and Y be nonempty subsets of a semigroup S. Then, the following properties hold:
1) If Y ⊆ X, then SXc⊆S˜ Yc.
2) SXc∩Se Yc = SXc∩Yc, SXc∪Se Yc = SXc∪Yc.
Proof . (1) is straightforward by Definition 3.1.10.
(2) Let s be any element of S. Suppose s ∈ Xc∩ Yc. Then, s ∈ Xc and s ∈ Yc.
Thus, we have
Suppose s /∈ Xc∩ Yc. Then, s /∈ Xc or s /∈ Yc. Hence, we have
(SXc∩Se Yc)(s) = SXc(s) ∩ SYc(s) = ∅ = SXc∩Yc(s)
Let s be any element of S. Suppose s ∈ Xc∪ Yc. Then, s ∈ Xc or s ∈ Yc. Thus,
we have
(SXc∪Se Yc)(s) = SXc(s) ∪ SYc(s) = U = SXc∪Yc(s)
Suppose s /∈ Xc∪ Yc. Then, s ∈ X and s ∈ Y . Hence, we have
(SXc∪Se Yc)(s) = SXc(s) ∪ SYc(s) = ∅ = SXc∪Yc(s)
3.2 Soft Union Semigroups
In this section, first we define soft union semigroups, then study their basic properties with respect to soft set operations and soft uni-product.
Definition 3.2.12. Let S be a semigroup and fS be a soft set over U . Then, fS is
called a soft union semigroup of S, if
fS(xy) ⊆ fS(x) ∪ fS(y)
for all x, y ∈ S. For the sake of brevity, soft union semigroup is abbreviated by SU -semigroup in what follows.
Example 3.2.13. Let S = {a, b, c, d} be the semigroup in Example 3.1.8 and fS
be a soft set over U = S3, symmetric group. If we construct a soft set such that
fS(a) = {(1)}, fS(b) = {(1), (123)}, fS(c) = {(1), (12), (123)}, fS(d) = {(1), (123)}
Now, let U = x 0 0 x | x ∈ Z3
, 2 × 2 matrices with Z3 terms, be the universal
set. We construct a soft set gS over U by
gS(a) = 0 0 0 0 , 1 0 0 1 , gS(b) = 0 0 0 0 , 1 0 0 1 , 2 0 0 2 , gS(c) = 1 0 0 1 , 2 0 0 2 , gS(d) = 0 0 0 0 , 2 0 2 0 . Then, since gS(dd) = gS(b) * gS(d) ∪ gS(d),
gS is not an SU -semigroup over U .
Note 3.2.14. It is easy to see that if fS(x) = ∅ for all x ∈ S, then fS is an
SU -semigroup over U . We denote such a kind of SU -semigroup by eθ, and called null soft union semigroup. It is obvious that eθ = SSc, i.e. eθ(x) = ∅ for all x ∈ S.
Lemma 3.2.15. Let fSbe any SU -semigroup over U . Then, we have the followings:
1) eθ ∗ eθ = eθ, fS∗ eθ⊇eeθ and eθ ∗ fS⊇eeθ. 2) fS∩eeθ = eθ and fS∪eeθ = fS.
Theorem 3.2.16. Let fS be a soft set over U . Then, fS is an SU -semigroup over
U if and only if
Proof . Assume that fS is an SU -semigroup over U . Let a ∈ S. If (fS∗ fS)(a) = U,
then it is obvious that
(fS∗ fS)(a) ⊇ fS(a), thus fS ∗ fS⊇fe S.
Otherwise, there exist elements x, y ∈ S such that a = xy. Then, since fS is an
SU -semigroup over U , we have:
(fS ∗ fS)(a) = \ a=xy (fS(x) ∪ fS(y)) ⊇ \ a=xy fS(xy) = \ a=xy fS(a) = fS(a) Thus, fS∗ fS⊇fe S.
Conversely, assume that fS∗ fS⊇fe S. Let x, y ∈ S and a = xy. Then, we have:
fS(xy) = fS(a) ⊆ (fS∗ fS)(a) = \ a=xy (fS(x) ∪ fS(y)) ⊆ fS(x) ∪ fS(y)
Hence, fS is an SU -semigroup over U . This completes the proof.
Theorem 3.2.17. A non-empty subset A of a semigroup of S is a subsemigroup of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ A, β, if x ∈ A is an SU -semigroup, where α, β ⊆ U such that α ⊇ β.
Proof . Suppose A is a subsemigroup of S and x, y ∈ S. If x, y ∈ A, then xy ∈ A. Hence, fS(xy) = fS(x) = fS(y) = β and so, fS(xy) ⊆ fS(x) ∪ fS(y). If x, y /∈ A,
then xy ∈ A or xy /∈ A. In any case, fS(xy) ⊆ fS(x) ∪ fS(y) = α. Thus, fS is an
SU -semigroup.
Conversely assume that fS is an SU -semigroup of S. Let x, y ∈ A. Then, fS(xy) ⊆
fS(x) ∪ fS(y) = β. This implies that fS(xy) = β. Hence, xy ∈ A and so A is a
subsemigroup of S.
Theorem 3.2.18. Let X be a nonempty subset of a semigroup S. Then, X is a subsemigroup of S if and only if SXc is an SU -semigroup of S.
Proof . Since SXc(x) = U, if x ∈ S \ X, ∅, if x ∈ X
and U ⊇ ∅, the rest of the proof follows from Theorem 3.2.17.
Proposition 3.2.19. Let fS and fT be SU -semigroup over U . Then, fS∨ fT is an
SU -semigroup over U .
Proof . Let (x1, y1), (x2, y2) ∈ S × T . Then,
fS∨T((x1, y1)(x2, y2)) = fS∨T(x1x2, y1y2)
= fS(x1x2) ∪ fT(y1y2)
⊆ (fS(x1) ∪ fS(x2)) ∪ (fT(y1) ∪ fT(y2))
= (fS(x1) ∪ fT(y1)) ∪ (fS(x2) ∪ fT(y2))
= fS∨T(x1, y1) ∪ fS∨T(x2, y2)
Therefore, fS∨ fT is an SU -semigroup over U .
Proposition 3.2.20. If fS and hS are SU -semigroups over U , then so is fS∪he S over U .
Proof . Let x, y ∈ S, then
(fS∪he S)(xy) = fS(xy) ∪ hS(xy)
⊆ (fS(x) ∪ fS(y)) ∪ (hS(x) ∪ hS(y))
= (fS(x) ∪ hS(x)) ∪ (fS(y) ∪ hS(y))
= (fS∪he S)(x) ∪ (fS∪he S)(y)
Therefore, fS∪he S is an SU -semigroup over U .
Proposition 3.2.21. Let fS be a soft set over U and α be a subset of U such
that α ∈ Im(fS), where Im(fS) = {α ⊆ U : fS(x) = α, f or x ∈ S}. If fS is an
SU -semigroup over U , then L(fS; α) is a subsemigroup of S.
Proof . Since fS(x) = α for some x ∈ S, then ∅ 6= L(fS; α) ⊆ S. Let x, y ∈ L(fS; α),
then fS(x) ⊆ α and fS(y) ⊆ α. We need to show that xy ∈ L(fS; α) for all
x, y ∈ L(fS; α). Since fS is an SU -semigroup over U , it follows that
fS(xy) ⊆ fS(x) ∪ fS(y) ⊆ α ∪ α = α
implying that xy ∈ L(fS; α). The proof is completed.
Definition 3.2.22. Let fS be an SU -semigroup over U . Then, the subsemigroups
L(fS; α) are called lower α-subsemigroups of fS.
Proposition 3.2.23. Let fSbe a soft set over U , L(fS; α) be lower α-subsemigroups
of fS for each α ⊆ U and Im(fS) be an ordered set by inclusion. Then, fS is an
SU -semigroup over U .
Proof . Let x, y ∈ S and fS(x) = α1 and fS(y) = α2. Suppose that α1 ⊆ α2. It is
obvious that x ∈ L(fS; α1) and y ∈ L(fS; α2). Since α1 ⊆ α2, x, y ∈ L(fS; α2) and
since L(fS; α) is a subsemigroup of S for all α ⊆ U , it follows that xy ∈ L(fS; α2).
Hence, fS(xy) ⊆ α2 = α1∪ α2 = fS(x) ∪ fS(y). Thus, fS is an SU -semigroup over
U .
Proposition 3.2.24. Let fS and fT be soft sets over U and Ψ be a semigroup
Proof . Let t1, t2 ∈ T . Since Ψ is surjective, then there exist s1, s2 ∈ S such that Ψ(s1) = t1 and Ψ(s2) = t2. Then, (Ψ?(f S))(t1t2) =T{fS(s) : s ∈ S, Ψ(s) = t1t2} =T{fS(s) : s ∈ S, s = Ψ−1(t1t2)} =T{fS(s) : s ∈ S, s = Ψ−1(Ψ(s1s2)) = s1s2} =T{fS(s1s2) : si ∈ S, Ψ(si) = ti, i = 1, 2} ⊆T{fS(s1) ∪ fS(s2) : si ∈ S, Ψ(si) = ti, i = 1, 2} = (T{fS(s1) : s1 ∈ S, Ψ(s1) = t1}) ∪ (T{fS(s2) : s2 ∈ S, Ψ(s2) = t2}) = (Ψ?(fS))(t1) ∪ (Ψ(fS))(t2)
Hence, Ψ(fS) is an SU -semigroup over U .
Proposition 3.2.25. Let fS and fT be soft sets over U and Ψ be a semigroup
homomorphism from S to T . If fT is an SU -semigroup over U , then so is Ψ−1(fT).
Proof . Let s1, s2 ∈ S. Then,
(Ψ−1(fT))(s1s2) = fT(Ψ(s1s2))
= fT(Ψ(s1)Ψ(s2))
⊆ fT(Ψ(s1)) ∪ fT(Ψ(s2))
= (Ψ−1(fT))(s1) ∪ (Ψ−1(fT))(s2)
In this chapter, we define soft union left ideals, right ideals, two-sided ideals, bi-ideals, interior ideals, quasi-ideals and generalized bi-ideals of semigroups and obtain their basic properties related with soft set operations and soft uni-product. Moreover, we give the interrelations of them in detail.
4.1 Soft Union Ideals of Semigroups
In this section, we define soft union left, right and two-sided ideals and obtain their basic properties related with soft set operations and soft uni-product.
Definition 4.1.26. A soft set fS over U is called a soft union left ideal of S over
U if fS(ab) ⊆ fS(b) and is called a soft union right ideal of S over if fS(ab) ⊆ fS(a)
for all a, b ∈ S. A soft set over U is called a soft union two-sided ideal or soft union ideal of S if it is both soft union left and soft union right ideal of S over U .
For the sake of brevity, soft union left and soft union right ideal is abbreviated by SU -left ideal and SU -right ideal, respectively.
Example 4.1.27. Consider the semigroup S = {0, x, 1} defined by the following table:
. 0 x 1
0 0 0 0
x 0 x x
1 0 x 1
Let fS be a soft set over S such that fS(0) = {0}, fS(1) = {0, 1, x}, fS(x) = {0, x}.
a soft set hS over S such that hS(0) = {0, 1}, hS(1) = {1}, hS(x) = {0, x, 1}, then,
hS(x1) = hS(x) * hS(1). Thus, hS is not an SU -left ideal of S and moreover since
hS(1x) = hS(x) * hS(1), hS is not an SU -right ideal of S over U .
Theorem 4.1.28. Let fS be a soft set over U . Then, fS is an SU -left ideal of S
over U if and only if
e
θ ∗ fS⊇fe S
Proof . First assume that fS is an SU -left ideal of S over U . Let s ∈ S. If
(eθ ∗ fS)(s) = U,
then it is clear that eθ ∗ fS⊇fe S. Otherwise, there exist elements x, y ∈ S such that
s = xy. Then, since fS is an SU -left ideal of S over U , we have:
(eθ ∗ fS)(s) = \ s=xy (eθ(x) ∪ fS(y)) ⊇ \ s=xy (∅ ∪ fS(xy)) = \ s=xy (fS(xy)) = fS(s) Thus, we have eθ ∗ fS⊇fe S.
Conversely, assume that eθ ∗ fS⊇fe S. Let x, y ∈ S and s = xy. Then, we have:
fS(xy) = fS(s) ⊆ (eθ ∗ fS)(s) = \ s=mn (eθ(m) ∪ fS(n)) ⊆ eθ(x) ∪ fS(y) = ∅ ∪ fS(y) = fS(y)
Hence, fS is an SU -left ideal over U . This completes the proof.
Theorem 4.1.29. Let fS be a soft set over U . Then, fS is an SU -right ideal of S
over U if and only if
fS ∗ eθ⊇fe S
Proof . Similar to the proof of Theorem 4.1.28.
Theorem 4.1.30. Let fS be a soft set over U . Then, fS is an SU -ideal of S over
U if and only if
fS∗ eθ⊇fe S and eθ ∗ fS⊇fe S
Corollary 4.1.31. eθ is both SU -right and SU -left ideal of S.
Theorem 4.1.32. A non-empty subset L of a semigroup of S is a left ideal of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ L, β, if x ∈ L
is an SU -left ideal of S, where α, β ⊆ U such that α ⊇ β.
Proof . Suppose L is a left ideal of S and x, y ∈ S. If y ∈ L, then xy ∈ L. Hence, fS(xy) = fS(y) = β. If y /∈ L, then xy ∈ L or xy /∈ L. In any case,
fS(xy) ⊆ fS(y) = α. Thus, fS is an SU -left ideal of S.
Conversely assume that fS is an SU -left ideal of S. Let y ∈ L and x ∈ S. Then,
fS(xy) ⊆ fS(y) = β. This implies that fS(xy) = β. Hence, xy ∈ L and so L is a
left ideal of S.
The above theorem is valid for SU -right ideals of S, too.
Theorem 4.1.33. Let X be a nonempty subset of a semigroup S. Then, X is a left ideal of S if and only if SXc is an SU -left ideal of S over U .
The above theorem is valid for SU -right and SU -ideals of S, too.
Proposition 4.1.34. Let fS be a soft set over U . Then, fS is an SU -ideal of S over
U if and only if
fS(xy) ⊆ fS(x) ∩ fS(y)
for all x, y ∈ S.
Proof . Let fS be an SU -ideal of S over U . Then,
fS(xy) ⊆ fS(x) and fS(xy) ⊆ fS(y)
for all x, y ∈ S. Thus, fS(xy) ⊆ fS(x) ∩ fS(y). Conversely, suppose that fS(xy) ⊆
fS(x) ∩ fS(y) for all x, y ∈ S. It follows that
fS(xy) ⊆ fS(x) ∩ fS(y) ⊆ fS(x) and fS(xy) ⊆ fS(x) ∩ fS(y) ⊆ fS(y).
Thus, fS is an SU -ideal of S over U .
It is obvious that every left (right, two-sided) ideal of S is a subsemigroup of S. Moreover, we have the following:
Theorem 4.1.35. Let fS be a soft set over U . If fS is an SU -left (right, two-sided)
ideal of S over U , then fS is an SU -semigroup over U .
Proof . We give the proof for SU -left ideals. Let fS be an SU -left ideal of S over
U . Then, fS(xy) ⊆ fS(y) for all x, y ∈ S. Thus, fS(xy) ⊆ fS(y) ⊆ fS(x) ∪ fS(y), so
fS is an SU -semigroup over U .
Proposition 4.1.36. If fS is an SU -right ideal of S over U , then
fS∩(ee θ ∗ fS)
Proof . Assume that fS is an SU -right ideal of S. Then,
e
θ ∗ (fS∩(ee θ ∗ fS)) = (eθ ∗ fS)∩(ee θ ∗ (eθ ∗ fS)) (by Theorem 3.1.9 (3)) = (eθ ∗ fS)∩((ee θ ∗ eθ) ∗ fS) (by Theorem 3.1.9 (1)) e
⊇ (eθ ∗ fS)∩(ee θ ∗ fS) (by Lemma 3.2.15 (1)) = eθ ∗ fS
e
⊇ fS∩(ee θ ∗ fS)
Thus, fS∩(ee θ ∗ fS) is an SU -left ideal of S over U . Also,
(fS∩(ee θ ∗ fS)) ∗ eθ = (fS∗ eθ)∩((ee θ ∗ fS) ∗ eθ) = (fS∗ eθ)∩(ee θ ∗ (fS∗ eθ)) e ⊇ (fS∗ eθ)∩(ee θ ∗ fS) (sincefS∗ eθ⊇fe S) e ⊇ fS∩(ee θ ∗ fS)
Hence, fS∩(ee θ ∗ fS) is an SU -right ideal of S over U . This completes the proof.
The following proposition can be proved similar to the above proposition, so we give the proposition without proof.
Proposition 4.1.37. If fS is an SU -left ideal of S over U , then
fS∩(fe S∗ eθ)
is an SU -ideal of S over U .
Theorem 4.1.38. Let fS be an SU -right ideal of S over U and gS be an SU -left
ideal of S over U . Then
Proof . Let fS and gS be SU -right and SU -left ideal of S over U , respectively.
Then, since fS, gS⊇eeθ always holds, we have:
fS∗ gS⊇fe S∗ eθ⊇fe S and fS∗ gS⊇eeθ ∗ gS⊇ge S
It follows that fS∗ gS⊇fe S∪ge S.
Now, we show that if fS is an SU -right ideal of S over U and gS is an SU -left ideal
of S over U , then
fS∗ gS*fe S e ∩gS
with the following example:
Example 4.1.39. Consider the semigroup S and SU -ideal fS in Example 4.1.27.
Let gS be a soft set over S such that gS(0) = {x, 1}, gS(x) = {x}, gS(1) = {x},
One can easily show that gS is an SU -ideal of S over U . However,
(fS∗ gS)(x) =
\
x=ab
(fS(a) ∪ gS(b)) = {0, 1, x} * (fS∩ge S)(x) = {x}.
Proposition 4.1.40. Let fS and hS be SU -left ideals of S over U . Then, fS ◦ hS
is an SU -left ideal of S over U .
Proof . Let fS and hS be SU -left ideal of S and x, y ∈ S. Then,
(fS∗ hS)(y) =
\
y=pq
(fS(p) ∪ hS(q))
If y = pq, then xy = x(pq) = (xp)q. Since fSis an SU -left ideal of S, fS(xp) ⊆ fS(p).
Thus, (fS ∗ hS)(y) = \ y=pq (fS(p) ∪ hS(q)) ⊇ \ xy=xpq (fS(xp) ∪ hS(q)) = (fS∗ hS)(xy)
So,
(fS ∗ hS)(xy) ⊆ (fS∗ hS)(y)
If y is not expressible as y = pq, then (fS ∗ hS)(y) = U ⊇ (fS ∗ hS)(xy). Thus,
fS∗ hS is an SU -left ideal of S.
The above proposition is valid for SU -right ideals of S, too. Now, we give the following propositions without proof. The proofs are similar to those in Section 3. Moreover, the below propositions are valid for SU -right ideals of S, too.
Proposition 4.1.41. Let fS and fT be SU -left ideals of S over U . Then, fS ∨ fT
is an SU -left ideal of S × T over U .
Proposition 4.1.42. If fS and hS are two SU -left ideals of S over U , then so is
fS∪he S of S over U .
Proposition 4.1.43. Let fS be a soft set over U and α be a subset of U such that
α ∈ Im(fS). If fS is an SU -left ideal of S over U , then L(fS; α) is a left ideal of S.
Definition 4.1.44. Let fS be an SU -left ideal of S over U . Then, the left ideals
L(fS; α) are called lower α-left ideals of fS.
Proposition 4.1.45. Let fS be a soft set over U , L(fS; α) be lower α-ideals of fS
for each α ⊆ U and Im(fS) be an ordered set by inclusion. Then, fS is an SU -left
ideal of S over U .
In order to show Proposition 4.1.43, we have the following example:
Example 4.1.46. Consider the semigroup in Example 3.1.8. Define a soft set fS over U = D2 = {e, x, y, yx} such that fS(a) = {x}, fS(b) = {e, x}, fS(c) =
{e, x, y}, fS(d) = {e, x, yx}. Then, one can easily show that fS is an SU -ideal of S
over U . By taking into account Im(fS), we have: L(fS; {x}) = {a}, L(fS; {e, x}) =
{a, b}, L(fS; {e, x, y}) = {a, b, c}, L(fS; {e, x, yx}) = {a, b, d} One can easily show
In order to show Proposition 4.1.45, we have the following example:
Example 4.1.47. Consider the semigroup in Example 3.1.8. Define a soft set fS over U = D2 = {e, x, y, yx} such that fS(a) = {e}, fS(b) = {e, y}, fS(c) =
{e, y, yx}, fS(d) = {e, x, y, yx}, By taking into account
Im(fS) = {{e}, {e, y}, {e, y, yx}, {e, x, y, yx}}
and considering that Im(fS) is ordered by inclusion, we have:
L(fS; α) = {a}, if α = {e} {a, b}, if α = {e, y} {a, b, c}, if α = {e, y, yx} {a, b, c, d}, if α = {e, x, y, yx}
Since {a}, {a, b}, {a, b, c} and {a, b, c, d} are two-sided ideals of S, fS is an SU -ideal
of S over U .
Now we define a soft set hSover U = D2such that hS(a) = {x}, hS(b) = {e, x, y, yx}, hS(c) =
{e, x}, hS(d) = {e, x, yx}. By taking into account Im(hS) is ordered by inclusion,
we have: L(hS; α) = {a}, if α = {x} {a, c}, if α = {e, x} {a, b, d}, if α = {e, x, yx} {a, b, c, d}, if α = {e, x, y, yx}
Since {a, c}S * {a, c} and S{a, c} * {a, c}, {a, c} is not a two-sided ideal of S. Moreover, since; hS(cc) = hS(b) * hS(c) hS is not an SU -ideal of S over U .
Proposition 4.1.48. Let fS and fT be soft sets over U and Ψ be a semigroup
isomorphism from S to T . If fS is an SU -left ideal of S over U , then so is Ψ?(fS)
Proposition 4.1.49. Let fS and fT be soft sets over U and Ψ be a semigroup
homomorphism from S to T . If fT is an SU -left ideal of T over U , then so is
Ψ−1(fT) of S over U .
4.2 Soft Union Bi-ideals of Semigroups
In this section, we define soft union bi-ideals and study their properties as regards soft set operations and soft uni-product.
Definition 4.2.50. An SU -semigroup fS over U is called a soft union bi-ideal of S
over U if
fS(xyz) ⊆ fS(x) ∪ fS(z)
for all x, y, z ∈ S.
For the sake of brevity, soft union bi-ideal is abbreviated by SU -bi-ideal in what follows.
Example 4.2.51. Let S = {0, a, b, c} be the semigroup with the operation table given below. + 0 a b c 0 0 0 0 0 a 0 a b 0 b 0 0 0 0 c 0 c 0 0
Define the soft set fS over U = Z5 such that fS(0) = {0}, fS(a) = {0, 1}, fS(b) =
{0, 3}, fS(c) = {0, 2}. Then, one can easily show that fS is an SU -bi-ideal of S over
Theorem 4.2.52. Let fS be a soft set over U . Then, fS is an SU -bi-ideal of S over
U if and only if
fS∗ fS⊇fe S and fS∗ eθ ∗ fS⊇fe S
Proof . First assume that fS is an SU -bi-ideal of S over U . Since fS is an
SU -semigroup over U , we have
fS ∗ fS⊇fe S.
Let s ∈ S. In the case, when (fS∗ eθ ∗ fS)(s) = U, then it is clear that fS∗ eθ ∗ fS⊇fe S,
Otherwise, there exist elements x, y, p, q ∈ S such that
s = xy and x = pq
Then, since fS is an SU -bi-ideal of S over U , we have:
fS(s) = fS(xy) = fS((pq)y) ⊆ fS(p) ∪ fS(y)
Thus, we have (fS∗ eθ ∗ fS)(s) = [(fS∗ eθ) ∗ fS](s) = \ s=xy [(fS∗ eθ)(x) ∪ fS(y)] = \ s=xy [(\ x=pq (fS(p) ∪ eθ(q)) ∪ fS(y)] = \ s=xy [(\ x=pq (fS(p) ∪ ∅) ∪ fS(y)] = \ s=pqy (fS(p) ∪ fS(y)) ⊇ \ s=pqy fS(pqy) = fS(xy) = fS(s)
Hence, fS ∗ eθ ∗ fS⊇fe S. Here, note that if x 6= pq, then (fS ∗ eθ)(x) = U and so
Conversely, assume that fS ∗ fS⊇fe S. Then, fS is an SU -semigroup of S. Let
x, y, z ∈ S and s = xyz. Since fS∗ eθ ∗ fS⊇fe S, we have
fS(xyz) = fS(s) ⊆ (fS∗ eθ ∗ fS)(s) = [(fS ∗ eθ) ∗ fS](s) = \ s=mn [(fS∗ eθ)(m) ∪ fS(n)] ⊆ (fS∗ eθ)(xy) ∪ fS(z) = [ \ xy=pq (fS(p) ∪ eθ(q)] ∪ fS(z) ⊆ ((fS(x) ∪ eθ(y)) ∪ fS(z) = fS(x) ∪ fS(z)
Thus, fS is an SU -bi-ideal of S over U . This completes the proof.
Theorem 4.2.53. A non-empty subset B of a semigroup of S is a bi-ideal of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ B, β, if x ∈ B is an SU -bi-ideal of S, where α, β ⊆ U such that α ⊇ β. Proof . Similar to Theorem 3.2.17.
Theorem 4.2.54. Let X be a nonempty subset of a semigroup S. Then, X is a bi-ideal of S if and only if SXc is an SU -bi-ideal of S over U .
Proof . It follows from Theorem 4.2.53.
It is known that every left (right, two-sided) ideal of a semigroup S is a bi-ideal of S. Moreover, we have the following:
Theorem 4.2.55. Every SU -left (right, two-sided) ideal of a semigroup S over U is an SU -bi-ideal of S over U .
Proof . Let fS be an SU -left (right, two-sided) ideal of S over U and x, y, z ∈ S.
Then, fS is as SU -semigroup. Moreover,
fS(xyz) = fS((xy)z) ⊆ fS(z) ⊆ fS(x) ∪ fS(z)
Thus, fS is an SU -bi-ideal of S.
Theorem 4.2.56. Let fS be any soft subset of a semigroup S and gS be any
SU -bi-ideal of S over U . Then, the soft uni-products fS ∗ gS and gS ∗ fS are
SU -bi-ideals of S over U .
Proof . We show the proof for fS∗ gS. To see that fS ∗ gS is an SU -bi-ideal of S
over U , first we need to show that fS∗ gS is an SU -semigroup over U . Thus,
(fS∗ gS) ∗ (fS∗ gS) = fS∗ (gS ∗ (fS∗ gS)) e ⊇ fS∗ (gS ∗ (eθ ∗ gS)) (since fS⊇eeθ) = fS∗ (gS ∗ eθ ∗ gS) e ⊇ fS∗ gS (since gS∗ eθ ∗ gS⊇ge S))
Hence, by Theorem 3.2.16, fS∗ gS is an SU -semigroup over U . Moreover we have:
(fS ∗ gS) ∗ eθ ∗ (fS∗ gS) = fS∗ (gS∗ (eθ ∗ fS) ∗ gS)
e
⊇ fS∗ (gS∗ eθ ∗ gS) (since eθ ∗ fS⊇eeθ) e
⊇ fS∗ gS
Thus, it follows that fS∗ gS is an SU -bi-ideal of S over U . It can be seen in a similar
way that gS∗ fS is an SU -bi-ideal of S over U . This completes the proof.
Proposition 4.2.57. Let fS and fT be SU -bi-ideals over U . Then, fS ∨ fT is an
SU -bi-ideal of S × T over U .
Proposition 4.2.58. If fS and hS are two SU -bi-ideals of S over U , then so is
Proposition 4.2.59. Let fS be a soft set over U and α be a subset of U such that
α ∈ Im(fS). If fS is an SU -bi-ideal of S over U , then L(fS; α) is a bi-ideal of S.
Definition 4.2.60. If fS is an SU -bi-ideal of S over U , then bi-ideals L(fS; α) are
called lower-α bi-ideals of fS.
Proposition 4.2.61. Let fS be a soft set over U , L(fS; α) be lower α bi-ideals
of fS for each α ⊆ U and Im(fS) be an ordered set by inclusion. Then, fS is an
SU -bi-ideal of S over U .
Proposition 4.2.62. Let fS and fT be soft sets over U and Ψ be a semigroup
isomorphism from S to T . If fS is an SU -bi-ideal of S over U , then so is Ψ?(fS) of
T over U .
Proposition 4.2.63. Let fS and fT be soft sets over U and Ψ be a semigroup
homomorphism from S to T . If fT is an SU -bi-ideal of T over U , then so is Ψ−1(fT)
of S over U .
4.3 Soft Union Interior Ideals of Semigroups
In this section, we define soft union interior ideals of semigroups, study their basic properties with respect to soft operations and soft union product.
Definition 4.3.64. Let fS be an SU -semigroup over U . Then, fS is called a soft
union interior ideal of S, if
fS(xay) ⊆ fS(a)
for all x, y, a ∈ S. For the sake of brevity, soft union interior ideal is abbreviated by SU -interior ideal in what follows.
Example 4.3.65. Consider the semigroup S = {a, b, c, d} with the following operation table:
. a b c d
a a a a a
b a a a a
c a a b a
d a a b b
Let U = D2 = {< x, y >: x2 = y2 = e, xy = yx} = {e, x, y, yx} be the universal set
and fS be soft set over U such that
fS(a) = {e}, fS(b) = {e, x, y}, fS(c) = {e, x}, fS(d) = {e, x, y}.
Then, one can easily show that fS is an SU -interior ideal over U .
Now, let U = S2 be the symmetric group. If we construct a soft set gS over U such
that
gS(a) = {(1), (12)}, gS(b) = {(1)}, gS(c) = {(1)}, gS(d) = {(12)}
then, since
gS(dcd) = gS(a) * gS(c),
gS is not an SU -interior ideal over U .
Note 4.3.66. It is easy to see that if fS(x) = ∅ for all x ∈ S, then fS is an
SU -interior ideal over U . We denote such a kind of SU -interior ideal by eθ. It is obvious that eθ = SSc, i.e. eθ(x) = ∅ for all x ∈ S.
Theorem 4.3.67. Let fS be a soft set over U . Then, fS is an SU -interior ideal
over U if and only if
e
θ ∗ fS∗ eθ⊇fe S
Proof . Assume that fS is an SU -interior ideal over U . Let a ∈ S. If (eθ ∗fS∗ eθ)(a) =
U, then it is obvious that
Otherwise, if there exist elements y, z, u and v of S such that x = yz and y = uv, then, since fS is an SU -interior ideal of S, we have
fS(x) = fS(yz) = fS(uvz) ⊆ fS(v). Thus, (eθ ∗ fS ∗ eθ)(x) = ((eθ ∗ fS) ∗ eθ)(x) = {\ x=yz (eθ ∗ fS)(y) ∪ eθ(z)} = \ x=yz {(\ y=uv (eθ(u) ∪ fS(v))) ∪ eθ(z)} = \ x=yz {(\ y=uv (∅ ∪ fS(v))) ∪ ∅} ⊇ \ x=yz {(\ y=uv (∅ ∪ fS(uvz))) ∪ ∅} = fS(x)
Thus, eθ ∗ fS∗ eθ⊇fe S. Note that if y 6= uv, then (eθ ∗ fS)(y) = U , and so (eθ ∗ fS∗ eθ)(x) =
U ⊇ fS(x).
Conversely, assume that eθ ∗ fS ∗ eθ⊇fe S. Let x, a, y be any element of S. Then, we
have: fS(xay) ⊆ (eθ ∗ fS∗ eθ)(xay) = \ xay=pq {(eθ ∗ fS)(p) ∪ eθ(q)} ⊆ (eθ ∗ fS)(xa) ∪ eθ(y) = (eθ ∗ fS)(xa) ∪ ∅ = \ xa=mn {eθ(m) ∪ fS(n)} ⊆ eθ(x) ∪ fS(a) = fS(a)
Hence, fS is an SU -interior ideal over U . This completes the proof.
Theorem 4.3.68. A non-empty subset I of a semigroup of S is an interior ideal of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ I, β, if x ∈ I is an SU -interior ideal, where α, β ⊆ U such that α ⊇ β.
Proof . Suppose I is an interior ideal of S and x, a, y ∈ S. If a ∈ I, then xay ∈ I. Hence, fS(xay) = fS(a) = β. If a /∈ I, then xay ∈ I or xay /∈ I. In any case,
fS(xay) ⊆ fS(a) = α. Thus, fS is an SU -interior ideal of S.
Conversely assume that fS is an SU -interior ideal of S. Let a ∈ I and x, y ∈ S.
Then, fS(xay) ⊆ fS(a) = β. This implies that fS(xay) = β. Hence, xay ∈ I and so
I is an interior ideal of S.
Theorem 4.3.69. Let X be a nonempty subset of a semigroup S. Then, X is an interior ideal of S if and only if SXc is an SU -interior ideal of S.
Proof . Since SXc(x) = U, if x ∈ S \ X, ∅, if x ∈ X
and U ⊇ ∅, the rest of the proof follows from Theorem 4.3.68.
It is obvious that every two-sided ideal of S is an interior ideal of S. Moreover, we have the following:
Proposition 4.3.70. Let fS be a soft set over U . If fS is an SU -ideal of S over U ,
then fS is an SU -interior ideal of S over U .
Proof . Let fS be an SU -ideal of S over U and x, y ∈ S. Then,
Hence, fS is an SU -interior ideal of S over U .
The following example shows that the converse of this property does not hold in general:
Example 4.3.71. Consider the SU -interior ideal fS in Example 4.3.65. Since
fS(cc) = fS(b) * fS(c)
fS is not an SU -ideal of S.
The converse of Proposition 4.3.70 holds for a regular semigroup as shown in the following theorem:
Theorem 4.3.72. Let fS be a soft set over U , where S is a regular semigroup.
Then, the following conditions are equivalent:
1) fS is an SU -ideal of S over U .
2) fS is an SU -interior ideal of S over U .
Proof . By Proposition 4.3.70, it suffices to prove that (2) implies (1). Assume that (2) holds. Let a, b be any elements of S. Then, since S is regular, there exist elements x and y in S such that
a = axa and b = byb.
Since fS is an SU -interior ideal of S, we have
fS(ab) = fS((axa)b) = fS((ax)a(b)) ⊆ fS(a),
and
fS(ab) = fS(a(byb)) = fS((a)b(yb)) ⊆ fS(b).
Proposition 4.3.73. Let S be a monoid and fS be a soft set over U . Then, fS is
an SU -ideal of S if and only if fS is an SU -interior ideal of S.
Proof . The necessity is clear by Proposition 4.3.70. Now let us show the sufficiency. For x, y ∈ S, fS(xy) = fS(xye) ⊆ fS(y) and fS(xy) = fS(exy) ⊆ fS(x). Thus, fS is
an SU -ideal of S.
It is known that a semigroup S is called left simple if it contains no proper left ideal of S, is called right simple if it contains no proper right ideal of S and is called simple if it contains no proper ideal.
Definition 4.3.74. A semigroup S is called soft union left simple if every SU -left ideal of S is a constant function, is called soft union right simple if every SU -right ideal of S is a constant function and is called soft union simple if every SU -ideal of S is a constant function.
Theorem 4.3.75. For a semigroup S, the following conditions are equivalent:
1) S is left simple.
2) S is soft union left simple.
Proof . First assume that S is left simple. Let fS be any SU -left ideal of S and a
and b be any element of S. Then, it follows from (Clifford and Preston, 1961, p. 6) that there exist elements x, y ∈ S such that b = xa and a = yb. Hence, since S is an SU -left ideal of S,
fS(a) = fS(yb) ⊆ fS(b) = fS(xa) ⊆ fS(a)
and so fS(a) = fS(b). Since a and b be any elements of S, this means that fS is a
constant function. Thus, we obtain that S is soft union left simple and (1) implies (2).
Conversely, assume that (2) holds. Let A be any left ideal of S. Then, SAc is an
of S. Then, since A 6= ∅,
SAc(x) = ∅
and so x ∈ A. This implies that S ⊆ A, and so S = A. Hence, S is left simple and (2) implies (1). In the case, when S is soft union right simple, the proof follows similarly.
Theorem 4.3.76. For a semigroup S, the following conditions are equivalent:
1) S is simple.
2) S is soft union simple.
Theorem 4.3.77. For a regular semigroup S, the following conditions are equivalent:
1) S is simple.
2) S is soft union simple.
3) Every SU -interior ideal of S is constant function.
Proof . The equivalence of (1) and (2) follows from Theorem 4.3.76. Assume that (2) holds. Let fS be any SU -interior ideal of S and a and b be any element of S.
Then, since S is simple, it follows from (Petrich, 1973, Lemma 1.3.9) that there exist elements x and y in S such that
a = xby. Then, since fS is an SU -interior ideal of S, we have
fS(a) = fS(xby) ⊆ fS(b).
One can similarly show that fS(b) ⊆ fS(a). Thus, fS(a) = fS(b). Since a and b
be any elements of S, fS is a constant function and so (2) implies (3). Since every
SU -interior ideal of S is an SU -ideal of S by the regularity of S, it is clear that (3) implies (2).
Definition 4.3.78. A soft set fS over U is called soft union semiprime if for all
a ∈ S,
fS(a) ⊆ fS(a2).
Proposition 4.3.79. Let fS be a soft union semiprime interior ideal of S. Then,
fS(an) ⊆ fS(an+1) for all positive integers n.
Proof . Let n be any positive integer. Then,
fS(an) ⊆ fS(a2n) ⊆ fS(a4n) = fS(a3n−2an+1a) ⊆ fS(an+1).
Definition 4.3.80. A semigroup S is called archimedean if for all a, b ∈ S, there exists a positive integer n such that an∈ SbS (Petrich, 1973).
Proposition 4.3.81. Let S be an archimedean semigroup. Then, every soft union semiprime interior ideal of S is a constant function.
Proof . Let fS be any soft union semiprime interior ideal of S and a, b any element
of S. Since S is archimedean, there exist elements x, y ∈ S such that
an = xby.
Thus, we have fS(a) ⊆ fS(an) = fS(xby) ⊆ fS(b). Similarly, we have fS(b) ⊆ fS(a)
and so fS(a) = fS(b). Since a and b be any elements of S, fS is a constant function.
Proposition 4.3.82. Let fS and fT be SU -interior ideals over U . Then, fS∨ fT is
an SU -interior ideal of S × T over U .
Proof . Let (x1, y1), (x2, y2), (x3, y2) ∈ S × T . Then,
fS∨T((x1, y1)(x2, y2)(x3, y3)) = fS∨T(x1x2x3, y1y2y3)
= fS(x1x2x3) ∪ fT(y1y2y3)
⊆ fS(x2) ∪ fT(y2)
= fS∨T(x2, y2)
Proposition 4.3.83. If fS and hS are SU -interior ideals of S over U , then so is
fS∪he S.
Proof . Let x, y, z ∈ S. Then, we have
(fS∪he S)(xyz) = fS(xyz) ∪ hS(xyz) ⊆ fS(y) ∪ hS(y)
= (fS∪he S)(y)
Therefore, fS∪he S is an SU -interior ideal of S over U .
Proposition 4.3.84. Let fS be a soft set over U and α be a subset of U such
that α ∈ Im(fS), where Im(fS) = {α ⊆ U : fS(x) = α, f or x ∈ S}. If fS is an
SU -interior ideal over U , then L(fS; α) is an interior ideal of S.
Proof . Since fS(x) = α for some x ∈ S, then ∅ 6= L(fS; α) ⊆ S. Let a ∈ L(fS; α)
and x, y ∈ S, then fS(a) ⊆ α. We need to show that xay ∈ L(fS; α) for all
a ∈ L(fS; α) and x, y ∈ S. Since fS is an SU -interior ideal of S over U , it follows
that
fS(xay) ⊆ fS(a) ⊆ α
implying that xay ∈ L(fS; α). Thus, the proof is completed.
Definition 4.3.85. Let fS be an SU -interior ideal over U . Then, the interior ideals
L(fS; α) are called lower α-interior ideals of fS.
Proposition 4.3.86. Let fS be a soft set over U , L(fS; α) be lower α-interior ideals
of fS for each α ⊆ U . Then, fS is an SU -interior ideal of S over U .
Proof . Let x, y ∈ S and fS(a) = α1 for some a ∈ S. It follows that a ∈ L(fS; α1).
Since L(fS; α) is an interior ideal of S for all α ⊆ U , it follows that xay ∈ L(fS; α1).
Hence, fS(xay) ⊆ α1 = fS(a). Thus, fS is an SU -interior ideal of S over U .
Example 4.3.87. Consider the semigroup in Example 4.3.65. Define a soft set fS
over U = Z such that fS(a) = {1}, fS(b) = {1, 3, 5}, fS(c) = {1, 2}, fS(d) = {1, 3}.
Then, one can easily show that fS is an SU -interior ideal of S over U . By taking
into account Im(fS), we have:
L(fS; {1}) = {a},
L(fS; {1, 2}) = {a, c},
L(fS; {1, 3}) = {a, d},
L(fS; {1, 3, 5}) = {a, b, d}
One can easily show that {a}, {a, c}, {a, d} and {a, b, d} are interior ideals of S.
In order to show Proposition 4.3.86, we have the following example:
Example 4.3.88. Consider the semigroup in Example 4.3.65. Define a soft set fS
over U = S3such that fS(a) = {(1)}, fS(b) = {(1), (12)}, fS(c) = {(1), (12), (13)}, fS(d) =
{(1), (12), (13), (123)}. By taking into account
Im(fS) = {{(1)}, {(1), (12)}, {(1), (12), (13)}, {(1), (12), (13), (123)}} we have: L(fS; α) = {a}, if α = {(1)} {a, b}, if α = {(1), (12)} {a, b, c}, if α = {(1), (12), (13)} {a, b, c, d}, if α = {(1), (12), (13), (123)}
Since {a}, {a, b}, {a, b, c} and {a, b, c, d} are interior ideals of S, fS is an SU -interior
ideal of S over U . Now we define a soft set hS over U = Z such that hS(a) =
By taking into account Im(fS) = {{−1, −2}, {−1, −2, −3}, {−1, −2, −4}, {−1, −2, −3, −4}} , we have L(hS; α) = {b}, if α = {−1, −2} {a, b}, if α = {−1, −2, −3} {b, d}, if α = {−1, −2, −4} {a, b, c, d}, if α = {−1, −2, −3, −4} Since S{b}S = {a} /∈ {b}, {b} is not an interior ideal of S. Moreover, since;
hS(bdc) = hS(a) * hS(d)
hS is not an SU -interior ideal of S over U .
Proposition 4.3.89. Let fS and fT be soft sets over U and Ψ be a semigroup
isomorphism from S to T . If fS is an SU -interior ideal of S over U , then Ψ?(fS) is
an SU -interior ideal of T over U .
Proof . Let t1, t2, t3 ∈ T . Since Ψ is surjective, then there exist s1, s2, s3 ∈ S such
that Ψ(s1) = t1, Ψ(s2) = t2, Ψ(s3) = t3. Then, (Ψ?(fS))(t1t2t3) =T{fS(s) : s ∈ S, Ψ(s) = t1t2t3} =T{fS(s) : s ∈ S, s = Ψ−1(t1t2t3)} =T{fS(s) : s ∈ S, s = Ψ−1(Ψ(s1s2s3)) = s1s2s3} =T{fS(s1s2s3) : si ∈ S, Ψ(si) = ti, i = 1, 2, 3} ⊆ (T{fS(s2) : s2 ∈ S, Ψ(s2) = t2}) = (Ψ?(fS))(t2)
Proposition 4.3.90. Let fS and fT be soft sets over U and Ψ be a semigroup
homomorphism from S to T . If fT is an SU -interior ideal of T over U , then Ψ−1(fT)
is an SU -interior ideal of S over U . Proof . Let s1, s2, s3 ∈ S. Then,
(Ψ−1(fT))(s1s2s3) = fT(Ψ(s1s2s3))
= fT(Ψ(s1)Ψ(s2)Ψ(s3))
⊆ fT(Ψ(s2))
= (Ψ−1(fT))(s2)
Hence, Ψ−1(fT) is an SU -interior ideal over U .
4.4 Soft Union Quasi-ideals of Semigroups
In this section, we define soft union quasi-ideals and study their properties as regards soft set operations, soft union product and certain kinds of soft union ideals. Definition 4.4.91. A soft set fS over U is called a soft union quasi-ideal of S over
U if
(fS∗ eθ)e∪(eθ ∗ fS)⊇fe S.
For the sake of brevity, soft union quasi-ideal is abbreviated by SU -quasi-ideal in what follows.
Proposition 4.4.92. Every SU -quasi-ideal of S is an SU -semigroup of S. Proof . Let fS be any SU -quasi-ideal of S. Since fS⊇eeθ,
fS∗ fS⊇eeθ ∗ fS and fS∗ fS⊇fe S∗ eθ.
Hence,
fS∗ fS⊇(ee θ ∗ fS)e∪(fS∗ eθ)⊇fe S
Proposition 4.4.93. Each one-sided SU -ideal of S is an SU -quasi-ideal of S. Proof . Let fS be an SU -left ideal of S. Since eθ ∗ fS⊇fe S, we have
(eθ ∗ fS)e∪(fS∗ eθ)e⊇eθ ∗ fS⊇fe S.
Thus, fS is an SU -quasi-ideal of S.
The converse of Proposition 4.4.93 does not hold in general as shown in the following example:
Example 4.4.94. Consider the semigroup S = {0, a, b, c} with the following operation table: . 0 a b c 0 0 0 0 0 a 0 a b 0 b 0 0 0 0 c 0 c 0 0
Let U = S3 be the universal set and fS be the soft set over U such that fS(0) =
{(12), (13)}, fS(a) = {(12), (13)}, fS(b) = {(1), (12), (13)}, fS(c) = {(1), (12), (13)}.
Then, one can easily show that fS is an SU -quasi-ideal of S, but since
fS(ca) = fS(c) * fS(a),
fS is not an SU -left ideal and so SU -ideal of S.
Proposition 4.4.95. Every SU -quasi-ideal of S is an SU -bi-ideal of S.
Proof . Let fS be an SU -quasi-ideal of S. Then, fS is an SU -semigroup. Moreover,
and so fS∗ eθ ∗ fS⊇(ee θ ∗ fS)e∪(fS∗ eθ)⊇fe S, as fS is an SU -quasi-ideal of S. Hence,
fS∗ eθ ∗ fS⊇fe S.
Thus, fS is an SU -bi-ideal of S.
The converse of Proposition 4.4.95 does not hold in general as shown in the following example:
Example 4.4.96. Consider the semigroup S = {0, 1, 2, 3} with the following table:
. 0 1 2 3
0 0 0 0 0
1 0 0 0 0
2 0 0 0 1
3 0 0 1 2
Let U = D3 = {< x, y >: x3 = y2 = e, xy = yx2} = {e, x, x2, y, yx, yx2} be the
universal set and fS be the soft set over U such that
fS(0) = {e}, fS(1) = {e, x, y}, fS(2) = {e, x}, fS(3) = {e, x, y, yx}.
Then, fS is an SU -bi-ideal of S. In fact, (fS ∗ fS)(0) = {e}, (fS ∗ fS)(1) =
{e, x, y, yx}, (fS ∗ fS)(2) = {e, x, y, yx}, (fS ∗ fS)(3) = U , and so fS ∗ fS⊇fe S.
Moreover, (fS ∗ eθ ∗ fS)(0) = {e}, (fS∗ eθ ∗ fS)(1) = {e, x, y, yx}, (fS∗ eθ ∗ fS)(2) =
U, (fS∗ eθ ∗ fS)(3) = U , and so fS∗ eθ ∗ fS⊇fe S.
However, fS is not an SU -quasi-ideal of S. In fact,
(fS∗ eθ)(1) = {e, x} (eθ ∗ fS)(1) = {e, x}
and so
, implying that fS is not an SU -quasi-ideal of S.
The converse of Proposition 4.4.95 holds for a regular semigroup as shown in the following sections.
Theorem 4.4.97. A non-empty subset Q of a semigroup of S is a quasi-ideal of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ Q, β, if x ∈ Q
is an SU -quasi-ideal of S, where α, β ⊆ U such that α ⊇ β.
Theorem 4.4.98. Let X be a nonempty subset of a semigroup S. Then, X is a quasi-ideal of S if and only if SXc is an SU -quasi-ideal of S over U .
Proof . It follows from Theorem 4.4.97.
Theorem 4.4.99. Let fS and gS be any SU -quasi-ideal of S over U . Then, the soft
union product fS∗ gS is an SU -bi-ideal of S over U .
Proof . Let fS be an SU -quasi-ideal of S. Then,
fS∗ eθ ∗ fS⊇eeθ ∗ eθ ∗ fS⊇eeθ ∗ fS and fS∗ eθ ∗ fS⊇fe S∗ eθ ∗ eθ⊇fe S∗ eθ
and so fS∗ eθ ∗ fS⊇(ee θ ∗ fS)e∪(fS∗ eθ)⊇fe S, as fS is an SU -quasi-ideal of S. Hence,
fS∗ eθ ∗ fS⊇fe S.
Then, we have
(fS∗ gS) ∗ (fS∗ gS) = (fS∗ gS∗ fS) ∗ gS⊇(fe S ∗ eθ ∗ fS) ∗ gS⊇fe S∗ gS
and (fS∗ gS) ∗ eθ ∗ (fS∗ gS) = (fS∗ (gS∗ eθ) ∗ fS) ∗ gS⊇(fe S∗ (eθ ∗ eθ) ∗ fS) ∗ gS⊇(fe S∗
e
Proposition 4.4.100. Let fS be any SU -right ideal of S and gS be any SU -left
ideal of S. Then, fS∪ge S is an SU -quasi-ideal of S.
Proof . Let fS be any SU -right ideal of S and gS be any SU -left ideal of S. Then,
((fS∪ge S) ∗ eθ)e∪(eθ ∗ (fS∪ge S))⊇(fe S∗ eθ)∪(ee θ ∗ gS)e⊇fS∪ge S.
Proposition 4.4.101. Let fS and gS be any SU -quasi-ideals of S. Then, fS∪ge S is
an SU -quasi-ideal of S.
Proof . Let fS and gS be any SU -quasi-ideals of S. Then,
((fS∪ge S) ∗ eθ)∪(ee θ ∗ (fS∪ge S))⊇(fe S∗ eθ)∪(ee θ ∗ fS)⊇fe S
and
((fS∪ge S) ∗ eθ)e∪(eθ ∗ (fSe∪gS))⊇(ge S∗ eθ)e∪(eθ ∗ gS)⊇ge S.
Thus,
((fS∪ge S) ∗ eθ)∪(ee θ ∗ (fS∪ge S))⊇fe S∪ge S.
Proposition 4.4.102. Let fS be a soft set over U and α be a subset of U such that
α ∈ Im(fS). If fS is an SU -quasi-ideal of S over U , then L(fS; α) is a quasi-ideal
of S.
Proof . Since fS(x) = α for some x ∈ S, then ∅ 6= L(fS; α) ⊆ S. Let a ∈ (S ·
L(fS; α) ∪ L(fS; α) · S). Then, there exist x, y ∈ L(fS; α) and s, r ∈ S such that
a = sx = yr.
Thus, fS(x) ⊆ α and fS(y) ⊆ α. Since
(eθ ∗ fS)(a) = { \ a=cd {eθ(c) ∪ fS(d)} ⊆ eθ(s) ∪ fS(x) = fS(x) ⊆ α
and (fS∗ eθ)(a) = { \ a=nm {fS(n) ∗ eθ(m)} ⊆ fS(y) ∪ eθ(r) = fS(y) ⊆ α
Since fS is an SU -quasi-ideal of S, we have
fS(a) ⊆ (eθ ∗ fS)(a) ∪ (fS∗ eθ)(a) ⊆ α,
thus a ∈ L(fS; α). This shows that L(fS; α) is a quasi-ideal of S.
Definition 4.4.103. Let fSbe an SU -quasi-ideal of S over U . Then, the quasi-ideals
L(fS; α) are called lower α-quasi-ideals of fS.
Theorem 4.4.104. Let S be a semigroup without zero. If S is a group, then every SU -quasi-ideal of S is a constant function.
Proof . Assume that S is a group and let fS be any SU -quasi-ideal of S and a and
b be any element of S. Then, since S is a group, there exist elements x and y of S such that
a = bx = yb. Since fS is an SU -quasi-ideal of S, we have
fS(a) ⊆ ((fS∗ eθ)∪(ee θ ∗ fS))(a)
= (fS∗ eθ)(a) ∪ (eθ ∗ fS)(a) = {\ a=pq fS(p) ∪ eθ(q)} ∪ { \ a=pq e θ(p) ∪ fS(q)} ⊆ (fS(b) ∪ eθ(x)) ∪ (eθ(y) ∪ fS(b)) = fS(b)
It can be seen in a similar way that fS(b) ⊆ fS(a). Thus, fS(a) = fS(b) for every
pair of elements a and b of S. This implies that fS is a constant function.
Proposition 4.4.105. Let fS be any SU -quasi-ideal of a commutative semigroup
S and a be any element of S. Then,
fS(an) ⊇ fS(an+1)
for every positive integer n.
Proof . For any positive integer n, we have
(fS ∗ eθ)(an+1) = \ an+1=xy (fS(x) ∪ eθ(y)) ⊆ fS(an) ∪ eθ(a) = fS(an). Similarly, (eθ ∗ fS)(an+1) ⊆ fS(an).
Thus, since fS is an SU -quasi-ideal of S
fS(an+1) ⊆ ((fS∗ eθ)∪(ee θ ∗ fS))(a
n+1)
= (fS ∗ eθ)(an+1) ∪ (eθ ∗ fS))(an+1)
⊆ fS(an) ∪ fS(an)
⊆ fS(an)
This completes the proof.
4.5 Soft Union Generalized Bi-ideals of Semigroups
In this section, we define soft union generalized bi-ideals and study their properties as regards soft set operations and soft union product.
Definition 4.5.106. A soft set over U is called a soft union generalized bi-ideal of S over U if
fS(xyz) ⊆ fS(x) ∪ fS(z)
for all x, y, z ∈ S. For the sake of brevity, soft union generalized bi-ideal is abbreviated by SU -generalized bi-ideal in what follows.
It is clear that every SU -bi-ideal of S is an SU -generalized bi-ideal of S, but the converse of this statement does not hold in general as shown in the following example: Example 4.5.107. Consider the semigroup S in Example 4.3.65. Define the soft set fS over U = Z4 such that fS(a) = {0}, fS(b) = {0, 1, 2}, fS(c) = {0, 1}, fS(d) =
{0, 1, 2}. Then, one can easily show that fS is an SU -generalized bi-ideal of S over
U . However since
fS(cc) = fS(b) * fS(c) ∪ fS(c)
fS is not an SU -bi-ideal of S.
However the converse of this theorem holds for a regular semigroup as shown in the following proposition:
Proposition 4.5.108. Every SU -generalized bi-ideal of a regular semigroup is an SU -bi-ideal of S.
Proof . Let fS be an SU -generalized bi-ideal of S and let a and b be any element of
S. Then, since S is regular, there exists an element x ∈ S such that b = bxb. Thus, we have
fS(ab) = fS(a(bxb)) = fS(a(bx)b) ⊆ fS(a) ∪ fS(b).
This implies that fS is an SU -semigroup of S, and so fS is an SU -bi-ideal of S.
The following theorems are given without proof. One can easily show them with the help of Section 4.2-Soft union bi-ideals of semigroups.
Theorem 4.5.109. Let fS be a soft set over U . Then, fS is an SU -generalized
bi-ideal of S over U if and only if
fS∗ eθ ∗ fS⊇fe S
Theorem 4.5.110. A non-empty subset G of a semigroup of S is a generalized bi-ideal of S if and only if the soft subset fS defined by
fS(x) = α, if x ∈ S \ G, β, if x ∈ G
is an SU -generalized bi-ideal, where α, β ⊆ U such that α ⊇ β.
Theorem 4.5.111. Let X be a nonempty subset of a semigroup S. Then, X is a generalized bi-ideal of S if and only if SXc is an SU -generalized bi-ideal of S over
U .
It is known that every SU -left (right, two-sided) ideal of a semigroup S over U is an SU -bi-ideal of S over U . Thus, we have the following:
Theorem 4.5.112. Every SU -left (right, two-sided) ideal of a semigroup S over U is an SU -generalized bi-ideal of S over U .
Theorem 4.5.113. Let fS be any soft subset of a semigroup S and gS be any
SU -bi-ideal of S over U . Then, the soft union products fS ∗ gS and gS ∗ fS are
SU -generalized bi-ideals of S over U .
Proposition 4.5.114. Let fS and fT be SU -generalized bi-ideals over U . Then,
fS∨ fT is an SU -generalized bi-ideal of S × T over U .
Proposition 4.5.115. If fS and hS are two SU -generalized bi-ideals of S over U ,
then so is fS∪he S of S over U .
Proposition 4.5.116. Let fS be a soft set over U and α be a subset of U such
that α ∈ Im(fS). If fS is an SU -generalized bi-ideal of S over U , then L(fS; α) is
