Lobachevskii Journal of Mathematics Vol. 11, 2002, 3-6 \begin{matrix}c \\ ◯ \end{matrix}M. Arslanov, N. Kehayopulu

M. Arslanov, N. Kehayopulu

A NOTE ON MINIMAL AND MAXIMAL IDEALS OF ORDERED SEMIGROUPS

Abstract. Ideals of ordered groupoids were defined by second author in [2]. Considering the question under what conditions an ordered semigroup (or semigroup) contains at most one maximal ideal we prove that in an ordered groupoid S without zero there is at most one minimal ideal which is the intersection of all ideals of S. In an ordered semigroup, for which there exists an element a ∈ S such that the ideal of S generated by a is S, there is at most one maximal ideal which is the union of all proper ideals of S. In ordered semigroups containing unit, there is at most one maximal ideal which is the union of all proper ideals of S.

We deal with groupoids without zero. If an ordered groupoid (respectively, a groupoid) S contains minimal ideals, equivalently, if the intersection of all ideals of S is non-empty, then the intersection of all ideals of S is the unique maximal ideal of S — the Kernel of S. So an ordered semigroup (groupoid or ordered groupoid) contains at most one minimal ideal. The following question is natural: under what conditions an ordered semigroup (or semigroup) contains at most one maximal ideal? Under what conditions an ordered semigroup (or semigroup) contains a unique maximal ideal? We prove the following: if S is an ordered semigroup (or semigroup) for which there is an element a ∈ S such that S ⊆ I(a) (equivalently, S = I(a)) (I(a) is the ideal of S generated by a), then any two maximal ideals of S coincide. If S is an ordered semigroup (or semigroup), a ∈ S and S ⊆ I(a), and if the set of all proper ideals of S is non-empty, then the union of all proper ideals of S is the unique maximal ideal of S. In an ordered semigroup (or semigroup) containing the unit element “e”, we have S ⊆ I(e). So in ordered semigroups (or semigroups) containing unit any two maximal ideals coincide. In an ordered semigroup (or semigroup) containing unit and proper ideals, the union of all proper ideals of S is the unique maximal ideal of S. If (S, ., ≤) is an ordered groupoid, a non-empty subset I of S is called a left (respectively, right) ideal of S [2] if

1) SI ⊆ I (respectively, IS ⊆ I); and

This is a part of the joint Research Program between Russia and Greece supported by the Ministry of Research and Technology, Bilatere Cooperation Division, Grant No. 70/3/4967.

2) a ∈ I, b ∈ S & b ≤ a implies b ∈ I. If (S, .) is a groupoid, a left (respectively, right) ideal of S is a non-empty subset I of S such that SI ⊆ I (respectively, IS ⊆ I). If S is a groupoid or an ordered groupoid and I both a left and a right ideals of S, then it is called an ideal of S. If (S, ., ≤) is an ordered groupoid, a zero of S is an element 0 of S such that 0x = x0 = 0 and 0 ≤ x for every x ∈ S [1]. If S is a groupoid, a zero of S is an element 0 of S such that 0x = x0 = 0 for every x ∈ S. Let S be a groupoid or an ordered groupoid without zero. An ideal M of S is called minimal if

For every ideal T of S such that T ⊆ M, we have T = M  (∗)

If S has a zero, then the ideal {0} satisfies condition (*). In fact we have the following: Let T be an ideal of S such that T ⊆ {0}. Since T is an ideal of S, we have 0 ∈ T, then {0} ⊆ T, and T = {0}. As a consequence, in groupoids (respectively, ordered groupoids) possessing a zero, the minimal ideal has to be defined as follows: Let S be a groupoid (respectively, ordered groupoid) with zero. An ideal M of S is called minimal if

1) M\{0}.

2) For every ideal T of S such that T ⊆ M, we have T = {0} or T = M. This is whatever the authors call 0-minimal ideal. So, when we speak about minimal ideals in groupoids or in ordered groupoids, we have to distinguish if the groupoids and the ordered groupoids have or do not have a zero.

1. In this section we consider groupoids and ordered groupoids without zero. Let S be an ordered groupoid. For H ⊆ S, we denote

(H] := {t ∈ S | t ≤ h for some h ∈ H}.

For convenience, we denote S^{1} := S ∪ {e} where e is a symbol such that ae = ea = a for all a ∈ S. Lemma 1. Let S be an ordered groupoid, L a left ideal and R a right ideal of S. Then the set (LR] is an ideal of S. Proof. We have ∅ ≠ (LR] ⊆ S (since L ≠ ∅, R ≠ ∅). S(LR] = (S](LR] ⊆ ((SL)R] ⊆ (LR], (LR]S = (LR](S] ⊆ (L(RS)] ⊆ (LR]. If a ∈ (LR] and S ∋ b ≤ a, then b ∈ (LR]. □ Therefore, if I, J are ideals of S, then the set (IJ] is an ideal of S. Lemma 2. Let S be a groupoid (ordered groupoid) and {I_{i} | i ∈ I} a family of ideals of S. If ∩{I_{i} | i ∈ I} ≠ ∅, then the set ∩{I_{i} | i ∈ I} is an ideal of S. Proposition 1. Let S be an ordered groupoid, and A be the set of all ideals of S. The following are equivalent:

1) S contains minimal ideals.

2) The set ∩{J | J ∈ A} is the only minimal ideal of S.

3) ∩{J | J ∈ A} ≠ ∅.

Proof. 1) ⇒ 2). Let I be a minimal ideal of S.

Then I = ∩{J | J ∈ A}. Indeed, let J be an ideal of S We have JI ⊆ SI ⊆ I, then (JI] ⊆ (I] = I. Since I is a minimal ideal and (JI] an ideal of S, we have I = (JI]. Since JI ⊆ JS ⊆ J, we have I = (JI] ⊆ (J] = J. Hence the set ∩{J | J ∈ A} is a minimal ideal of S.

2) ⇒ 3). Since ⊆ {J | J ∈ A} is an ideal of S, we have ∩{J | J ∈ A} ≠ ∅.

3) ⇒ 1). Since ∩{J | J ∈ A} ≠ ∅, by Lemma 2, the set ∩{J | J ∈ A} is an ideal of S. The set ∩{J | J ∈ A} is a minimal ideal of S. Indeed, let K be an ideal of S such that K ⊆ ∩{J | J ∈ A}. Since K is an ideal of S, we have ∩{J | J ∈ A} ⊆ K. □ Therefore, if an ordered groupoid S contains minimal ideals, equivalently, if the intersection of all ideals of S is non-empty, then this minimal ideal is unique, and it is the intersection of all ideals of S. This is the so-called Kernel of S. Note that Proposition 1 also holds for groupoids instead of ordered groupoids.

2. In this section we consider maximal ideals of ordered semigroups. An ideal M of a groupoid or an ordered groupoid S is called proper if M ≠ S [3]. Since there is no ideal T of S such that T ⊃ S, exactly as in the case of ordered groupoids with zero, the maximal ideal has to be defined (only for proper ideals) as follows: a proper ideal M of a groupoid or an ordered groupoid S is called maximal if for each ideal T of S such that T ⊇ M, we have T = M or T = S (equivalently, there is no ideal T of S such that M ⊂ T ⊂ S) (cf. also [2]). If S is an ordered semigroup (respectively, semigroup) and ∅ ≠ A ⊆ S, we denote by I(A) the ideal of S generated by A i. e. the smallest – under inclusion relation– ideal of S containing A. For an ordered semigroup S, we have I(A) = (A ∪ SA ∪ AS ∪ SAS] (cf. [2]). For a semigroup S, we have I(A) = A ∪ SA ∪ AS ∪ SAS. For A = {a}, we write I(a) instead of I({a}). Lemma 3. Let (S, ., ≤) be an ordered semigroup, a ∈ S and S ⊆ I(a). Let M_{1} be a proper ideal of S and M_{2} a maximal ideal of S. Then M_{1} ⊆ M_{2}. Proof. Let x ∈ M_{1} such that x ∉ M_{2}. We have

M_{2} ⊆ M_{2} ∪ {x} ⊆ I(M_{2} ∪ {x}).

Since I(M_{2} ∪ {x}) is an ideal and M_{2} a maximal ideal of S, we have

I(M_{2} ∪ {x}) = M_{2} or I(M_{2} ∪ {x}) = S.

Since x ∈ M_{2} ∪ {x} ⊆ I(M_{2} ∪ {x}) and x ∉ M_{2}, we have I(M_{2} ∪ {x}) ≠ M_{2}. Thus I(M_{2} ∪ {x}) = S. On the other hand,

I(M_{2} ∪ {x}) = ((M_{2} ∪ {x}) ∪ S(M_{2} ∪ {x}) ∪ (M_{2} ∪ {x})S ∪ S(M_{2} ∪ {x})S].

Hence we have

a ∈ ((M_{2} ∪ {x}) ∪ S(M_{2} ∪ {x}) ∪ (M_{2} ∪ {x})S ∪ S(M_{2} ∪ {x})S].

Then there exist u, v ∈ S^{1} and y ∈ M_{2} ∪ {x} such that a ≤ uyv. If y ∈ M_{2}, then uyv ∈ S^{1}M_{2}S^{1} ⊆ M_{2}, a ∈ M_{2}, S ⊆ I(a) ⊆ M_{2}, and M_{2} = S. A contradiction. If y = x, then a ≤ uxv ∈ S^{1}M_{1}S^{1} ⊆ M_{1}, a ∈ M_{1}, S ⊆ I(a) ⊆ M_{1}, and M_{1} = S. Again this is a contradiction. □

As an immediate consequence of Lemma 3, we have the following Lemma 4. Let S be an ordered semigroup, a ∈ S, and S ⊆ I(a). Then, for any two maximal ideals M_{1} and M_{2} of S, we have M_{1} = M_{2}. Lemma 5. Let S be a groupoid (resp. ordered groupoid) and {I_{i} | i ∈ I} a non-empty family of ideals of S. Then the set ∪{I_{I} | i ∈ I} is an ideal of S. Proposition 2. Let S be an ordered semigroup, a ∈ S, S ⊆ I(a). Let A be the set of all proper ideals of S. If A ≠ ∅, then the set ∪{J | J ∈ A} is the unique maximal ideal of S. Proof. Since A ≠ ∅, then by Lemma 5 the set ∪{J | J ∈ A} is an ideal of S. The set ∪{J | J ∈ A} is a proper ideal of S. Indeed, let ∪{J | J ∈ A} = S. Let J ∈ A such that a ∈ J. Since J is an ideal of S, a ∈ J, and S ⊆ I(a), we have S ⊆ I(a) ⊆ J, then J = S. This is impossible. Let now T be an ideal of S such that ∪{J | J ∈ A} ⊆ T, and let T ≠ S. Since T ∈ A, we have T ⊆ ∪{J | J ∈ A}. Then T = ∪{J | J ∈ A}. If M is a maximal ideal of S then, by Lemma 4, we have M = ∪{J | J ∈ A}. So the set ∪{J | J ∈ A} is the unique maximal ideal of S. □ Therefore, if an ordered semigroup for which there is an element a ∈ S such that S ⊆ I(a) contains proper ideals, then the union of all proper ideals of S is an ideal, and the union of all proper ideals of S is the unique maximal ideal of S. Remark. If S is an ordered semigroup containing the unit element “e”, then we have S = Se ⊆ (Se] ⊆ I(e). □ Now it follows from Lemma 3, Lemma 4 and Proposition 2 the following: Lemma 6. Let S be an ordered semigroup containing unit. Let M_{1} be a proper ideal of S, and M_{2} be a maximal ideal of S. Then M_{1} ⊆ M_{2}. Lemma 7. Let S be an ordered semigroup containing unit. Then for any two maximal ideals M_{1} and M_{2} of S, we have M_{1} = M_{2}. Proposition 3. Let S be an ordered semigroup containing unit. If the set A of all proper ideals of S is non-empty, then the set ∪{J | J ∈ A} is the unique maximal ideal of S. □

References

[1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll, Publ. XXV, Providence, 1967.

[2] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Mathematica Japonica 35, No. 6 (1990), 1051–1056.

[3] N. Kehayopulu, Note on Green’s relations in ordered semigroups, Mathematica Japonica 36, No. 2 (1991), 211–214

Marat Arslanov, Kazan State University, Department of Mathematics and Mechanics, Kremlevskaya, 18, Kazan: 420008, Russia E-mail address: Marat.Arslanov@ksu.ru Niovi Kehayopulu, University of Athens, Department of Mathematics, Nikomidias 18, 161 22 Kesariani, Greece E-mail address: nkehayop@cc.uoa.gr

Received October 20, 2002