Archimedean ordered semigroups as ideal extensions Niovi Kehayopulu · Michael Tsingelis Communicated by Mikhail V. Volkov Abstract The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Keywords Ideal extension of an ordered semigroup by an ordered semigroup · Archimedean ordered semigroup · Idempotent element · Ideal · Simple ordered semigroup · Nil ordered semigroup · The Rees quotient ordered semigroup.
1. Introduction For an ordered semigroup S and H ⊆ S, denote (H] := {t ∈ S | t ≤ h for some h ∈ H} [10]. Throughout the paper we denote by N = {1, 2, . . . } the set of natural numbers. A nonempty subset A of S is called a left (resp. right) ideal of S if (1) SA ⊆ A (resp. AS ⊆ A) and (2) If a ∈ A and b ∈ S, b ≤ a, then b ∈ A. N. Kehayopulu University of Athens, Department of Mathematics, 15784 Panepistimiopolis, Athens, Greece e-mail:
[email protected] M. Tsingelis Department of Electrological Engineering, Technological Educational Institute of Patras, Megalou Alexandrou 1, 26334 Koukouli, Patras, Greece e-mail:
[email protected]
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Further, A is called an ideal of S if it is both a left and a right ideal of S [10]. For an element a ∈ S, we denote by I(a) the ideal of S generated by a, i.e. the least (with respect to the inclusion relation) ideal of S containing a. We have I(a) = (a ∪ Sa ∪ aS ∪ SaS] [10]. Using the standard notation S 1 = S ∪ {1}, where 1 is a symbol such that 1a = a = a1 for each a ∈ S 1 , we write I(a) = (S 1 aS 1 ]. There are several definitions of archimedean ordered semigroups in the bibliography, such as the definitions given by Fuchs [8], Birkhoff [2], Saito [22], Tamura and Kimura [23] etc., with slight differences between them, introduced for different purposes and mostly inspired by ordered groups, ordered rings (cf. also [8,2,1]). The concept of archimedean ordered semigroup introduced by Kehayopulu et al. in [14] was completely different from the definitions previously used by other authors. It has been inspired by the concept of archimedean semigroups given by Petrich [19] and has answered to a problem posed by Kehayopulu in [9]. Since then many papers on ordered semigroups based on that definition appeared showing the importance of the concept and its applications to semigroups (without order) (cf., for example, the papers by Cao [3] and Xie [24]). The definition of archimedean semigroup given by Petrich is the following: A semigroup S is called archimedean if for any a, b ∈ S there exists an n ∈ N such that an ∈ SbS [19]. The definition of archimedean ordered semigroups given by Kehayopulu et al. in [14] was the following: for each a, b ∈ S there exists n ∈ N such that an ≤ xby for some x, y ∈ S. Clearly, an ordered semigroup is archimedean if and only if for each a, b ∈ S there exists n ∈ N such that an ∈ (SbS] (cf. also [15]). As we have seen in [17], the same concept can be equivalently defined as follows: an ordered semigroup S is archimedean if for any a, b ∈ S there exists n ∈ N such that bn ∈ I(a). This last definition being an extended form of the definition of an archimedean semigroup considered by Tamura and Kimura in [23] and Putcha in [20,21] shows that the definitions of archimedean semigroups considered by Petrich, Tamura, Kimura, and Putcha are actually the same. It might be finally noted that if (S, .) is an archimedean semigroup in their sense, then endowing it with the equality relation ≤= {(x, y) | x = y} yields an ordered semigroup (S, ., ≤) which is archimedean as an ordered semigroup in our sense. This means that our concept of an archimedean ordered semigroup generalizes the corresponding concept for semigroups without order. For ideal extensions of semigroups (without order) we refer to [6,7,19]. For ideal extensions of ordered semigroups we refer to [16]. For characterization of archimedean semigroups (without order) containing an idempotent as ideal extensions of simple semigroups containing an idempotent 2
by nil semigroups we refer to [5,18]. In the present paper we characterize the archimedean ordered semigroups containing an idempotent by means of ideal extensions. For the Rees quotient we use the notation S|K instead of the usual one S/K. This is because using expressions like S/K and S\K at the same time might cause confusion. We prove that if S is an archimedean ordered semigroup and e an idempotent element of S, then there exists an ideal K of S containing e such that K is a simple subsemigroup of S and the Rees quotient S|K is nil. This means that each archimedean ordered semigroup containing an idempotent is an ideal extension of a simple ordered semigroup by a nil ordered semigroup. Conversely, we prove that if K is an ideal and at the same time a simple subsemigroup of an ordered semigroup S and the Rees quotient ordered semigroup S|K is nil, then S is archimedean. From this it follows that if an ordered semigroup is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then it is archimedean. Thus, we characterize the archimedean ordered semigroups which contain an idempotent as ideal extensions of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Let (S, ., ≤) be an ordered semigroup. The zero of S is an element of S, usually denoted by 0, such that 0 ≤ x and 0x = x0 = 0 for all x ∈ S [2]. An idempotent of S is an element e of S such that e2 = e. Let (S, ., ≤S ), (T, ∗, ≤T ) be ordered semigroups, f : S → T a mapping from S into T . The mapping f is called isotone if x, y ∈ S, x ≤S y implies f (x) ≤T f (y) and reverse isotone if x, y ∈ S, f (x) ≤T f (y) implies x ≤S y. (Observe that each reverse isotone mapping is one-to-one). The mapping f is called a homomorphism if it is isotone and satisfies f (xy) = f (x) ∗ f (y) for all x, y ∈ S. The mapping f is called an isomorphism if it is a reverse isotone onto homomorphism. The ordered semigroups S and T are called isomorphic if there exists an isomorphism between them.
2. Main results Definition 1 [14,15]. An ordered semigroup S is called archimedean if for each a, b ∈ S there exists n ∈ N such that an ∈ (SbS]. Definition 2 [11,12]. An ordered semigroup S is called simple if S is the only ideal of S. That is, if A is an ideal of S, then A = S. A subsemigroup K of an ordered semigroup (S, ., ≤) is called simple if the ordered semigroup (K, ., ≤) is simple (that is, if K with the multiplication and the order of S is a simple ordered semigroup). 3
Definition 3 [4,13]. An element a of an ordered semigroup S having a zero 0 is called nilpotent if there exists an n ∈ N such that an = 0. An ordered semigroup S having a zero is called nil if every element of S is nilpotent, that is, for each a ∈ S there exists n ∈ N such that an = 0. Definition 4 [16]. An ordered semigroup V is called an ideal extension (or just an extension) of an ordered semigroup S by an ordered semigroup Q, if Q has a zero 0, S ∩ (Q\{0}) = ∅, and there exists an ideal K of V such that K ≈ S and V |K ≈ Q. Lemma 1. [16] Let (S, ., ≤) be an ordered semigroup and K an ideal of S. Let S|K := (S\K) ∪ {0}, where 0 is an arbitrary element of K (S\K is the complement of K to S). We define an operation ” ∗ ” and an order ” 4 ” on S|K as follows: ∗ : S|K × S|K → S|K | (x, y) → x ∗ y where ½ xy if xy ∈ S\K x ∗ y := 0 if xy ∈ K 4:= (≤ ∩[(S\K) × (S\K)]) ∪ {(0, x) | x ∈ S|K}. Then (S|K, ∗ , 4) is an ordered semigroup and 0 is its zero. The following observation is obvious. Lemma 2. Let S be an ordered semigroup, K an ideal of S and a ∈ S\K. Then we have the following: (i) aρ ∈ S\K =⇒ aρ = a | ∗ a ∗{z. . . ∗ a} (ii) aρ ∈ K
for all ρ ∈ N.
ρ
=⇒ a | ∗ a ∗{z. . . ∗ a} = 0
for all ρ ∈ N, ρ ≥ 2.
ρ
Lemma 3. If S is an ordered semigroup, then (SaS] is an ideal of S for every a ∈ S. Proof. Let a ∈ S. Then (SaS] 6= ∅ because a3 belongs to (SaS]. S(SaS] = (S](SaS] ⊆ (S 2 aS] ⊆ (SaS] and (SaS]S = (SaS](S] ⊆ (SaS 2 ] ⊆ (SaS]. Clearly, if x ∈ (SaS] and y ∈ S such that y ≤ x, then y ∈ (SaS]. This immediately implies Lemma 4. Let S be an ordered semigroup. Then S is simple if and only if S = (SaS] for every a ∈ S. 4
Theorem 1. Let (S, ., ≤) be an ordered semigroup. If S is archimedean and e is an idempotent element of S, then there exists an ideal K of S containing e such that K is a simple subsemigroup of S and the ordered semigroup S|K is nil. Conversely, let K be an ideal of S such that K is a simple subsemigroup of S and the ordered semigroup S|K be nil. Then S is archimedean. Proof. Suppose S is an archimedean ordered semigroup and e an idempotent element of S. Let K := (SeS]. Then we have the following: (1) K is an ideal of S and e ∈ K. Indeed, by Lemma 3, K is an ideal of S. Since e = e3 ∈ SeS ⊆ (SeS] = K, we have e ∈ K. (2) K is a simple subsemigroup of S. For this we have to check that K = (KbK] for any b ∈ K. In other words, for any a, b ∈ K, we have to prove that there exist x, y ∈ K such that a ≤ xby. Indeed, since a ∈ K = (SeS], we have a ≤ set for some s, t ∈ S. Since e is idempotent, we have a ≤ (se)e(et). Since S is archimedean and e, b ∈ S, there exists n ∈ N such that e = en ∈ (SbS]. Then e ≤ wbz for some w, z ∈ S. Hence we have a ≤ (se)e(et) ≤ (se)(wbz)(et) = (sew)b(zet). Since s, ew ∈ S, we get sew = se(ew) ∈ SeS ⊆ (SeS] = K. Similarly we obtain zet ∈ K. (3) The ordered semigroup S|K is nil, that is, for any x ∈ S|K, there exists n ∈ N such that x | ∗ x ∗{z· · · ∗ x} = 0. Indeed, since x ∈ S|K, we have n
x ∈ S\K or x = 0. For x = 0 the claim is true. Let x ∈ S\K. Since S is archimedean and x, e ∈ S, there exists k ∈ N such that xk ≤ aeb for some a, b ∈ S. Clearly ae, eb ∈ S. Since xk ∈ S and xk ≤ aeb = (ae)e(eb) ∈ SeS, we have xk ∈ (SeS] = K. If k = 1, then x ∈ K which is impossible. So we have k ≥ 2. Since x ∈ S\K and xk ∈ K (k ∈ N, k ≥ 2), by Lemma 2(ii), we have x | ∗ x ∗{z· · · ∗ x} = 0. k
The converse statement: Suppose K is an ideal of S which is a simple subsemigroup of S and the ordered semigroup S|K is nil. Then we have the following: (1) For each a ∈ S there exists m ∈ N such that am ∈ K. Indeed, let a ∈ S. If a ∈ K, then condition (1) is satisfied. Let a ∈ S\K. Since a ∈ S|K and S|K is nil, there exists h ∈ N such that |a ∗ a ∗{z· · · ∗ a} = 0. On the other h
hand, ah ∈ S\K or ah ∈ K. Let ah ∈ S\K. Since a ∈ S\K and ah ∈ S\K, 5
by Lemma 2(i), we have ah = a | ∗ a ∗{z. . . ∗ a} = 0 ∈ K which is impossible. h
Thus we have ah ∈ K. (2) S is archimedean. Let a, b ∈ S. Then there exists m ∈ S such that am ∈ (SbS]. Indeed, by (1), there exist m, k ∈ N such that am , bk ∈ K. Since K is a simple semigroup, by Lemma 4, we have am ∈ (Kbk K]. If k = 1, then am ∈ (KbK] ⊆ (SbS]. If k ≥ 2, then am ∈ (Kbk K] = (Kbbk−1 K] ⊆ (SbS]. Remark. If S is an ordered semigroup and K an ideal of S, then S is an ideal extension of K by S|K. Clearly K ∩ [(S|K)\{0}] = ∅, K ≈ K and S|K ≈ S|K under the identity mapping. By Theorem 1 and the above Remark, we immediately have the following Theorem 2. Let S be an ordered semigroup. If S is archimedean and contains an idempotent, then S is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. Corollary. An ordered semigroup S is archimedean and contains an idempotent if and only if S is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Acknowledgement. The authors would like to thank Prof. M. V. Volkov for valuable discussions and the anonymous referee for a number of useful remarks.
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