ORDERED SEMIGROUPS HAVING THE P -PROPERTY. Niovi Kehayopulu, Michael Tsingelis

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1 ORDERED SEMIGROUPS HAVING THE P -PROPERTY Niovi Kehayopulu, Michael Tsingelis ABSTRACT. The main results of the paper are the following: The ordered semigroups which have the P -property are decomposable into archimedean semigroups. Moreover, the ordered semigroups which have the P -property are decomposable into semigroups having the P -property. Conversely, if an ordered semigroup S is a complete semilattice of semigroups which have the P -property, then S itself has the P - property as well. An ordered semigroup is CS-indecomposable and has the P -property if and only if it is archimedean. If S is an ordered semigroup, then the relation N := {(a, b) N(a) = N(b)} (where N(a) is the filter of S generated by a (a S)) is the least complete semilattice congruence on S and the class (a) N is CS-indecomposable subsemigroup of S for every a S. The concept of the P m -property is introduced and a characterization of the P m -property in terms of the P -property is given. Our methodology simplifies the proofs of the corresponding results of semigroups (without order). 1. INTRODUCTION-PREREQUISITES If S be a semigroup and a, b S, we say that a divides b, if b I(a) (where I(a) is the principal ideal of S generated by a (a S)). A semigroup S is said to be archimedean if for every pair (a, b) of elements in S each one divides a power of the other [2]. Chrislock proved that if S is medial, that is, it satisfies the identity xyzu = xzyu, then S is a semilattice of archimedean semigroups Mathematics Subject Classification. 06F05. Key words and phrases. Archimedean ordered semigroup, P -property, complete semilattice of semigroups of type T, ideal, filter, CS-indecomposable ordered semigroup, P m -property. 1

2 The following question was natural: What is the biggest class of semigroups in which S is a semilattice of archimedean semigroups? A complete solution to that problem has been given by Putcha in [12]. A congruence ρ on a semigroup S is called semilattice congruence (S-congruence) if S/ρ is a semilattice. A semigroup S is called S-indecomposable if S cannot be homomorphic onto a semilattice L except the trivial one, L = 1. Tamura defines the P -property (power-property) of S as follows: If a, b S, b I(a), then there exists a natural number k such that b k I(a). One can find the following results by Tamura in [13]: (1) A semigroup S is a semilattice of archimedean semigroups if and only if S has the P -property. (2) If S is a semigroup, then there exists a smallest semilattice congruence ρ 0 on S and each ρ 0 -class is S-indecomposable. (3) A semigroup S has the P -property if and only if it is a semilattice of semigroups having the P - property. (4) A semigroup S is S-indecomposable and has the P -property if and only if it is archimedean. He gives a direct proof of the main part, the if part of (4) without using the general argument like the one mentioned in (2) as it was treated in the commutative or medial case. Furthermore, for a natural number m, the P m -property has been introduced and its characterization in terms of the P -property is given in [13]. In the present paper we extend the results on semigroups mentioned above for ordered semigroups. Our methodology gives more simplified proofs of the corresponding results on semigroups (without order) which can be also obtained as an easy modification of the results on ordered semigroups given in this paper. Let S be an ordered semigroup. For a subset H if S, we denote by (H] the subset of S defined by (H] := {t S t h for some h H}. 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 S b a, then b A. A is called an ideal of S if it is both a left and a right ideal of S [4]. If T is a subsemigroup of S, we denote by I T (a) the ideal of T generated by a, i.e. the least (with respect to the inclusion relation) ideal of T containing a. For T = S, we write I(a) instead of I S (a). For each a S, we have I(a) = (a Sa as SaS], L(a) = (a Sa] R(a) = (a as] [4]. A subsemigroup F of S is called a filter of S [3] if (1) a, b S 2

3 and ab F imply a F and b F and (2) a F and F c a imply c F. We denote by N(x) the filter of S generated by x (x S), and by N the relation on S defined by: N := {(x, y) S S N(x) = N(y)} [5]. An equivalence relation σ on an ordered semigroup (S,., ) is called a left (resp. right) congruence on S if (x, y) σ implies (zx, zy) σ (resp. (xz, yz) σ) for every z S. If σ is both left and a right congruence on S, then it is called a congruence on S. A semilattice congruence is a congruence σ on S such that (x 2, x) σ and (xy, yx) σ for all x, y S [5]. A semilattice congruence σ on S is called complete if x y implies (x, xy) σ [7]. A complete semilattice congruence σ on an ordered semigroup (S,., ) can be equivalently defined as an equivalence relation σ on S having the following properties: (1) (x, y) σ (xz, yz) σ z S (2) (xy, yx) σ x, y S (3) x y (x, xy) σ [10]. If σ is a semilattice congruence on S then, for each x S, the class (x) σ is a subsemigroup of S. The ordered semigroup S is called a complete semilattice of semigroups of a given type, say T, if there exists a semilattice congruence σ on S such that the class (x) σ is a subsemigroup of S of type T for every x S. Equivalent Definition: There exists a complete semilattice Y and a family {S α α Y } of archimedean subsemigroups of S such that: (1) S α S β = for every α, β Y, α β (2) S = S α α Υ (3) S α S β S αβ for every α, β Y. (4) S β (S α ] implies β α, where is the order of the semilattice Y defined by := {(α, β) α = αβ (= βα)} [9]. For convenience, we use the notation S 1 := S {1}, where 1 / S, 1a := a, a1 := a for each a S, 1.1:=1. We denote by N := {1, 2,... } the set of natural numbers. 2. MAIN RESULTS An ordered semigroup S is said to be archimedean if for each a, b S there exists k N such that b k (SaS] [6,8]. 3

4 Proposition 1. An ordered semigroup S is archimedean if and only if for every a, b S there exists k N such that b k I(a). Proof. =. Let a, b S. Then, there exists k N such that b k (SaS] I(a). =. Let a, b S. By hypothesis,there exist a, b S and k N such that b k I(a). Then b k zat for some z, t S 1. Then b k+2 b(zat)b = (bz)a(tb). Since b S, z S 1, we have bz S. Since t S 1, b S, we have tb S. Since (bz)a(tb) SaS, we have b k+2 (SaS]. Definition 2. Let S be an ordered semigroup. We say that S has the P-property if the following assertion is satisfied: a, b S, b I(a) = b k I(a 2 ) for some k N. Remark 3. We have a k I(b) for some k N if and only if there exists m N such that a m (SbS]. Thus the P -property can be equivalently defined as follows: a, b S, b I(a) = b k (Sa 2 S] for some k N. Remark 4. If S is an archimedean ordered semigroup, then for each a, b S there exists k N such that b k I(a 2 ). By Remark 4, we have the following: Corollary 5. Each archimedean ordered semigroup has the P -property. Lemma 6. [10; Theorem 2.8] An ordered semigroup has the P -property if and only if it is a complete semilattice of archimedean semigroups. Theorem 7. An ordered semigroup S has the P -property if and only if it is a complete semilattice of semigroups which have the P -property. Proof. =. Since S has the P -property, by Lemma 6, it is a complete semilattice of archimedean semigroups. Then, by Corollary 5, S is a complete semilattice of semigroups having the P -property. =. Let σ be a complete semilattice congruence on S such that (x) σ has the P -property for every x S. Let a, b S, b I(a). Suppose x, y S 1 such that b xay. Then, since σ is complete, we have (b, bxay) σ, so (bxay, b) σ ( ) On the other hand, since b 2 xayb, we have b 8 = b 2 b 2 b 2 b 2 (xayb)(xayb)(xayb)(xayb) = (xaybx)(aybxa)(ybxayb) ( ) 4

5 Since σ is a semilattice congruence on S, we have (xaybx, aybx 2 ) σ. Since (x 2, x) σ, we have (aybx 2, aybx) σ. Then, since (aybx, bxay) σ, we have (xaybx, bxay) σ. Then, by ( ), (xaybx, b) σ, and xaybx (b) σ. In a similar way we get aybxa (b) σ and ybxayb (b) σ. Then, by ( ), we have b 8 I (b)σ (aybxa). Since b 8, aybxa (b) σ, b 8 I (b)σ (aybxa), and the subsemigroup (b) σ of S has the P -property, by Remark 3, there exists k N such that (b 8 ) k ((b) σ (aybxa) 2 (b) σ ]. Then (b 8 ) k z(aybxa) 2 w = z(aybxa)(aybxa)w = (zaybx)a 2 (ybxaw) for some z, w (b) 1 σ. Since a, b S and x, y, z, w S 1, we have zaybx S, ybxaw S. Then, for the element m := 8k N, we have b m I(a 2 ), thus S has the P -property. An ordered semigroup S is called CS-indecomposable if S S is the unique complete semilattice congruence on S (i.e. if σ is a complete semilattice congruence on S, then σ = S S) [11]. Proposition 8. If S is an ordered semigroup, then we have the following: (1) N is the least complete semilattice congruence on S. (2) The class (a) N is CS-indecomposable subsemigroup of S for every a S. For the proof of (1) we refer to [7] and for (2) to [11]. Remark 9. (a) By Proposition 8, each ordered semigroup is a complete semilattice of CS-indecomposable semigroups. (b) The only if part of Lemma 6 is a consequence of Theorem 7 as well. Theorem 10. An ordered semigroup is CS-indecomposable and has the P - property if and only if it is archimedean. Proof. =. Since S has the P -property, by Lemma 6, it is a complete semilattice of archimedean semigroups. Let σ be a complete semilattice congruence on S such that (x) σ is an archimedean subsemigroup of S for every x S. Let a S (S ). Since S is CS-indecomposable, we have σ = S S, then (a) σ = S, and 5

6 S is archimedean. =. Since S is archimedean, by Corollary 5, S has the P -property. Let now σ be a complete semilattice congruence on S. Then σ = S S. In fact: Let a, b S. Since S is archimedean, there exist k N such that b k xay for some x, y S. Since b N(b), we have b k N(b). Since S xay b k N(b), we have xay N(b), then a N(b), and N(a) N(b). By symmetry, we get N(b) N(a), thus N(a) = N(b), and (a, b) N. Clearly N S S, so N = S S. By Proposition 8(1), N is the least complete semilattice congruence on S, that is N σ. Then S S σ, besides σ S S, so σ = S S. Definition 11. Let S be an ordered semigroup and m N. S has the P m - property if the following assertion is satisfied: a, b S, b I(a) = b k I(a m ) for some k N. Equivalently, a, b S, b I(a) = b k (Sa m S] for some k N. Remark 12. The following are obvious: (a) Every ordered semigroup has the P 1 -property. (b) The P -property and the P 2 -property are the same. Lemma 13. [10; Proposition 2.4] If an ordered semigroup has the P -property, then it also has the P m -property for every m N. Theorem 14. Let S be an ordered semigroup having the P m -property for some m N, m 2. Then S has the P λ -property for every λ N. Proof. Let λ N and a, b S, b I(a). Then there exists k N such that b k I(a λ ). In fact: Since S has the P m -property, there exists ν N such that b ν I(a m ). The following assertion is satisfied: For each n N there exists k n N such that b k n I(a mn ). ( ) Indeed: For n = 1 and for the element k 1 := ν N, we have b k 1 = b ν I(a m ) = I(a m1 ). Suppose property ( ) is true for the element n N. That is, there exists k n N such that b kn I(a mn ). Then there exists k n+1 N such that b k n+1 I(a mn+1 ). Indeed: Since b k n, a mn S (k n, m, n N), b k n I(a mn ) and S has the P m -property, there exists l N such that b lk n = (b k n ) l I((a mn ) m ) = I(a mn+1 ), where k + 1 := lk n N. Thus condition ( ) is satisfied. 6

7 Let now n 0 N such that m n0 > λ (such an n 0 exists). By ( ), there exists k n0 N such that b k n 0 I(a m n 0 ). Since m n0 > λ, we have m n0 λ N, then a mn 0 = a mn 0 λ + λ = a mn0 λ a λ Sa λ I(a λ ). Hence we have b kn 0 I(a m n 0 ) I(a λ ), where k n0 N, and the proof is complete. By Lemma 13, Theorem 14 and Remark 12(b), we have the following: Corollary 15. Let S be an ordered semigroup. The following are equivalent: (1) S has the P -property. (2) S has the P m -property for each m N. (3) S has the P m -property for some m N, m 2. References [1] J. L. Chrislock, On medial semigroups, J. Algebra 12 (1969), 1 9. [2] A. H. Clifford, G. B. Preston, The Algebraic Theory of Semigroups, Amer. Math. Soc., Math. Surveys 7, Providence, Rhode Island, [3] N. Kehayopulu, On weakly commutative poe-semigroups, Semigroup Forum 34, No. 3 (1987), [4] N. Kehayopulu, On weakly prime ideals of ordered semigroups, Mathematica Japonica 35, No. 6 (1990), [5] N. Kehayopulu, Remark on ordered semigroups, Mathematica Japonica 35, No. 6 (1990), [6] N. Kehayopulu, P. Kiriakuli, S. Hanumantha Rao, P. Lakshmi, On weakly commutative poe-semigroups, Semigroup Forum 41, No. 3 (1990), [7] N. Kehayopulu, M. Tsingelis, Remark on ordered semigroups. In : E. S. Ljapin, (edit.), Decompositions and Homomorphic Mappings of Semigroups, Interuniversitary collection of scientific works, St. Petersburg: Obrazovanie (1992), (ISBN ). [8] N. Kehayopulu, M. Tsingelis, On weakly commutative ordered semigroups, Semigroup Forum 56, No. 1 (1998),

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