ON STABILITY IN SEPARATIVE SEMIGROUP Abstract Stability, as introduced by Koch and Wallace (1956), has long stood as a central notion in semigroup theory, ensuring that inclusions of principal ideals collapse into equalities and that left and right structures align harmoniously. Separativity, on the other hand, generalises cancellativity while retaining algebraic regularity, and Burmistrovich's decomposition theorem revealed that every separative semigroup can be expressed as a semilattice of cancellative semigroups. Yet, whether separ- ative semigroups inherit stability in the sense of Koch and Wallace has remained unresolved. In this work, we close this gap: we prove that semilattices of can- cellative semigroups are stable, and hence every separative semigroup is inherently stable. This result elevates stability from a supplementary condition to a built-in feature of separative semigroups, o�ering a uni�ed perspective that strengthens the foundations of semigroup theory and deepens its structural coherence. Keywords: Stability in semigroup; Separative Semigroup; Cancellative semigroup; Semil- latice decomposition; and Green's Relations. Otobong J. Tom∗ Fed. Uni. Tech., Ikot Abasi & Akwa Ibom State Uni., Ikot Akpaden Otobong G. Udoaka Department of Mathematics, Akwa Ibom State University, Ikot Akpaden IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 1 Ekere S. Udo�a Department of Mathematics, Akwa Ibom State University, Ikot Akpaden mailto: tomdgreatest@gmail.com otobongawasi@aksu.edu.ng ekereudofia@yahoo.com 1 Introduction The concept of stability in semigroups was formally introduced by Koch and Wallace [16] in their seminal paper. A semigroup is said to be stable if for all a, b ∈ S: aS ⊆ abS =⇒ aS = abS, Sa ⊆ Sab =⇒ Sa = Sab. This algebraic de�nition re�ects a form of �ideal stability� and has since become a cor- nerstone in the theory of semigroups, inspiring further structural investigations [1, 12]. Stability is important because it guarantees that the behaviour of principal ideals remains robust under multiplication. The study of stability has evolved, particularly through its connection with Green's relations, which classify semigroup elements based on their di- visibility properties [16]. A crucial result from Koch and Wallace states that in a stable semigroup, Green's D− and J− equivalences coincide, implying that left and right ideal structures behave symmetrically. Anderson, et al [11] further expanded on this idea by studying stability conditions in various semigroup classes, highlighting its role in deter- mining structural simplicity and regularity. The structure of quasi-separative semigroup was introduced by Drazin [12], where the connections between it and other semigroup properties, such as inversity, regularity, etc, were established. This was later extended by Krasilnikova and Novikove [22]. Parallel to this development, the notion of separativity was introduced to generalise cancellativity while preserving a degree of regularity. East and Higgings [9] explored Green's relation in greater depth, providing a more re�ned classi�cation of semigroup elements' stability constraints. Hewitt and Zuckerman [6] applied separativity properties in their study, titled �l1− algebra of a commutative semi- group�, where they introduce harmonic analy sis on discrete commutative semigroups. Shourijeh [2] established the commutativity of separative semigroup and discussed its left presentation. Burmistrovich [7] provided a powerful structural result: a semigroup is separative if and only if it is isomorphic to a semilattice of cancellative semigroups. This theorem places separative semigroups in a broader algebraic context and has been exten- sively used in decomposition theory [13]. However, Burmistrovich's theorem itself does not discuss stability, leaving a gap in the understanding of whether separative semigroups inherit stability (in the sense of [16]). This gap motivates the present study. We aim to establish that separative semigroups are indeed stable in the sense of [19]. Our approach hinges on re-examining Burmistrovich's decomposition theorem from the perspective of stability. We �rst establish that a semilattice of cancellative semigroups is stable�a fact that has not been explicitly recorded in the literature. Since separative semigroups are precisely those semigroups isomorphic to such semilattices, it follows directly that every separative semigroup is stable. 2 Preliminaries De�nition 2.1 (Groupoid). Let S be a non-empty set. Let ∗ be an operation such that ∗ : S×S → S be de�ned on S. Then (S, ∗) is called groupoid if for all a, b ∈ S, a∗ b ∈ S. De�nition 2.2 (Semigroup). A groupoid is called semigroup if the binary operation is associative (i.e., for all a, b, c ∈ S, we have (a ∗ b) ∗ c = a ∗ (b ∗ c)). De�nition 2.3 (Idempotent). An element e ∈ S is called an idempotent element if e2 = e. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 2 De�nition 2.4 (Cancellative semigroup). A semigroup is called cancellative for all a, b, c ∈ S, ab = ac =⇒ a = b(left-cancellative) and ba = ca (right-cancellative). For more about this, the reader is referred to [1,4,5, 7]. 3 Separative semigroup De�nition 3.1. A semigroup S is called separative if and only if the following two conditions hold for all a, b ∈ S: � a2 = ab and ba = b2 together imply a = b, and � a2 = ba and ab = b2 together imply a = b. The reader can also see [15] De�nition 3.2 (Qusi-separative). A semigroup is called quasi�separative if a2 = ab = ba = b2 =⇒ a = b, [22]. On a bright-line, one can say that a separative semigroup is a quasi-separative. Ex- tracting from [20] we de�ne a weakly separative semigroup when we have asa = asb = bsa = bsb only if a = b for all a, b, s ∈ S. Theorem 3.1 (Proposition 1 of [12]). If S is any quasi-separative semigroup, then, for all a, b ∈ S we have a2 = ab = b2 if and only if a = b. And the converse also holds. Proof. If a2 = ab = b2, then we have (ab)2 = a2b2 = (aa)(bb) = a2b2, and (ab)(ba) = (aa)a(a) = a4, (ba)(ab) = (bb)b(b) = b4. Also, (ba)2 = b(ab)a = (bb)(aa) = b2a2, and a(ab) = a(aa)a = a3, b(ba) = b(bb)b = b3. Therefore, (ab)(ab) = (ba)(ba) = (ab)(ba), thus, by quasi-separativity, we have ab = ba. So we obtain a2 = ab = ba = b2, and by quasi-separativity again we have a = b. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 3 4 Semilattice of a Semigroup De�nition 4.1 (Semilattice). A semilattice is a commutative idempotent semigroup. That is, a set Y with a binary operation ∧ : Y × Y → Y such that, for all e, f, g ∈ Y : 1. (e ∧ f) ∧ g = e ∧ (f ∧ g) (Associativity), 2. e ∧ f = f ∧ e (Commutativity), 3. e ∧ e = e (Idempotency). Interpretation: � The operation ∧ can be thought of as a �meet� (greatest lower bound) in an ordered set. � The partial order associated with a semilattice is given by e ≤ f ⇐⇒ e ∧ f = e. � Thus, a semilattice is both an algebraic structure (a special semigroup) and an order-theoretic one (a meet-semilattice) in a poset. De�nition 4.2 (Semilattice of Semigroups). Let Y be a semilattice with operation ∧. A semilattice of semigroups is a semigroup S together with a decomposition S = ⋃ e∈Y Se, where each Se is a subsemigroup of S, such that for all e, f ∈ Y : Se · Sf ⊆ Se∧f . That is, the product of an element from Se and an element from Sf always lies in the same component corresponding to the meet e ∧ f , see �gure 1 below. Se Sf Se∧f a ∈ Se b ∈ Sf ab ∈ Se∧f Figure 1: Product in a semilattice of semigroups: a ∈ Se, b ∈ Sf multiply into Se∧f . Remarks: IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 4 � Each component Se is itself a semigroup. � The semilattice Y controls how the components interact with each other: multipli- cation �descends� to the meet in Y . � If all components Se are cancellative semigroups, then S is called a semilattice of cancellative semigroups. De�nition 4.3 (Green's Relations). Let S be a semigroup. Green's relations are the equivalence relations L,R,J ,H,D de�ned on S as follows: � The L-relation: For a, b ∈ S, aL b ⇐⇒ S1a = S1b, that is, a and b generate the same principal left ideal. Here S1 denotes S with identity adjoined if necessary. � The R-relation: For a, b ∈ S, a R b ⇐⇒ aS1 = bS1, that is, a and b generate the same principal right ideal. � The J -relation: For a, b ∈ S, a J b ⇐⇒ S1aS1 = S1bS1, that is, a and b generate the same two-sided ideal. � The H-relation: H = L ∩R, that is, a H b if and only if aL b and aR b. � The D-relation: D = L ◦ R = R ◦ L, that is, a D b if and only if there exists c ∈ S such that a L c and c R b. De�nition 4.4 (Associated Preorders). Besides the equivalence relations, one often con- siders the following preorders: � The ≤L-relation: for a, b ∈ S, a ≤L b ⇐⇒ S1a ⊆ S1b. � The ≤R-relation: for a, b ∈ S, a ≤R b ⇐⇒ aS1 ⊆ bS1. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 5 � The ≤J -relation: for a, b ∈ S, a ≤J b ⇐⇒ S1aS1 ⊆ S1bS1. For more clearity, reader should see [3, 8, 10,17,19�21] Theorem 4.1. Let S be a semigroup. Then D = J if and only if for every a, b ∈ S, SaS = SbS ⇒ ∃x ∈ S such that aLxR b. Proof. (⇒): Suppose D = J . Let a, b ∈ S with SaS = SbS, i.e. aJ b. Since D = J , this implies aD b. By the de�nition of D, there exists x ∈ S such that aLx and xR b. (⇐): Suppose the stated condition holds. Take any a, b ∈ S with aJ b, i.e. SaS = SbS. By hypothesis, there exists x ∈ S with aLxR b. Hence aD b. Since a, b ∈ S were arbitrary, this shows J ⊆ D. But in any semigroup it is always true that D ⊆ J (see [1, 18,19]). Therefore, D = J . 5 Results De�nition 5.1 (Koch�Wallace Stability [16]). A semigroup S is called stable if for all a, b ∈ S: 1. (Right stability): aS ⊆ abS =⇒ aS = abS, and 2. (Left stability): Sa ⊆ Sab =⇒ Sa = Sab. This de�nition was also given by East using Green's relation in [9]. Equivalently, if a ≤J ab, then aRab, and if a ≤J ba, then aLba. It is now necessary to establish some useful relationship between stable and separative semigroups. This is achieved through the following Propositions and theorems. Proposition 5.1. Let S = ⋃ e∈Y Se be a semilattice (with meet ∧) of cancellative semi- groups Se, so that SeSf ⊆ S e∧f for all e, f ∈ Y. Then S is separative. Proof. Let a, b ∈ S, suppose a2 = ab and ba = b2. Let a ∈ Se, b ∈ Sf . Since a 2 ∈ Se but ab ∈ Se∧f , the equality a 2 = ab forces e = e∧ f and so e ≤ f . Likewise, b2 ∈ Sf and ba ∈ Se∧f then b2 = ba forces f ≤ e. Hence, e = f , and so a, b lie in the same cancellative component Se. From a2 = ab, we have aa = ab =⇒ a = b. hus, the �rst separative implication holds. The second (symmetric) implication is identical in structure. Therefore, S satis�es the separativity conditions and so is separative. Theorem 5.1. Let S = ⋃ e∈Y Se be a semilattice (index set Y with meet ∧) of cancellative semigroups Se. If a ∈ Se and b ∈ Sf then ab ∈ Se∧f . Then S is stable, i.e. for all a, b ∈ S: IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 6 1. If Sa ⊆ S(ab) then Sa = S(ab). 2. If aS ⊆ (ab)S then aS = (ab)S. Proof. Write principal left ideals by Sa = {xa : x ∈ S} and right ideals by aS = {ax : x ∈ S}. Let a ∈ Se and b ∈ Sf . (1) Assume Sa ⊆ S(ab). In particular a ∈ S(ab), so there exists t ∈ S with a = t(ab). Writing t ∈ Sg, the product t(ab) ∈ Sg∧e∧f , while a ∈ Se. Equality of components forces e = g ∧ e ∧ f , hence e ≤ f and so e ∧ f = e. Thus ab ∈ Se. For any h ∈ Y , the map φh : Sh∧ea→ Sh∧e(ab) de�ned by φh(xa) = (xa)b is bijective by cancellativity. Taking unions over h yields Sa = S(ab). (2) Symmetrically, assume aS ⊆ (ab)S. Then a = (ab)u for some u ∈ S. This forces e ≤ f and ab ∈ Se. For each h, the map ψh : aSe∧h → (ab)Se∧h, ψh(ax) = (ab)x, is bijective. Taking unions gives aS = (ab)S. Hence S is stable. Remark 5.1. The proof uses (i) semilattice decomposition of separative semigroups, and (ii) cancellativity in each component, ensuring injectivity of the maps xa 7→ (xa)b and ax 7→ (ab)x. These yield stability (in the sense of [16]). Theorem 5.2. (Burmistrovich's Theorem [10]). A semigroup S is separative if and only if it is isomorphic to a semilattice of cancellative semigroups. Proof. See [7] 6 Conclusion In this work, we have established that every separative semigroup is stable. The argu- ment relies on two fundamental ingredients: �rst, the structural theorem of Burmistro- vich (Theorem 5.2), which asserts that every separative semigroup is isomorphic to a semilattice of cancellative semigroups; second, the fact that semilattices of cancellative semigroups preserve the stability conditions (Theorem 5.1). By combining these facts along with Proposition 5.1, we conclude that the separative property not only encodes a re�ned form of cancellativity but also guarantees algebraic stability in the sense of Koch and Wallace. This result strengthens the conceptual link between structural decompo- sition and stability theory in semigroup theory. It shows that separativity, originally introduced to capture subtle cancellation phenomena, naturally enforces stability (in the sense of Koch and Wallace) through its semilattice decomposition. Consequently, the class of separative semigroups provides a robust and stable framework for further explo- ration in the algebraic theory of semigroups. Declaration of Interest: 1. Funding: Not applicable 2. Informed Consent Statement: Not applicable 3. Data Availability: Not applicable 4. Con�ict of Interest Statement: No con�ict of interest. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 7 References [1] A. H. Cli�ord and G. B. Preston, The Algebraic Theory of Semigroups, American Mathematical Society (1961). [2] B. T. Shourijeh, C∗−Algebras of Some Semigroups, Honam Mathematical J. vol. 26, No. 4. pp. 483�507, 2004. [3] C. Hollings, Mathematics across the Iron Curtain: A History of Soviet Mathematics, American Mathematical Society, 2014. [4] D. D. Miller and Cli�ord, Regular classes in Semigroups, Tran. Amer. Math. Soc. vol. 82, 1956. [5] D. Rees, On Semi-Groups, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 36, no. 4, pp. 387�400 1940. [6] E. Hewitt, H. S. Zuckerman, The L1˘algebra of commutative semigroup, Trans. Amer. Math. Soc. , 83, pp. 70-97, 1956. [7] I. E. Burmistrovitch, Commutative bands of cancellative semigroups, Siberian Mat. Z. vol. 6, pp. 284�299, 1965. [8] J.A. Green, On the structure of semigroups, Ann. of Math., pp. 163-�172, 1951. [9] J. East and P. M. Higgins, Green's relations and stability for subsemigroups, Semi- group Forum, vol. 101, No. 1, pp. 77�86, 2020. [10] O. G. Udoaka, Generators and Inner Automorphism, The Colloquium-A Multidisci- plinary Thematc Policy Journal vol. 10, No. 1, pp. 102�111, 2022. [11] L. W. Anderson, R. P. Hunter and R. J. Koch, Some Results on Stability in Semi- groups. AMS Journal, pp. 521�529, 1965. [12] M. P. Drazin, A Partial Order in Completely Regular Semigroups, Journal of Algebra, vol. 98, pp. 362�374, 1986. [13] M. Petrich, Introduction to Semigroups, Charles Babbage Research Centre, 1984. [14] N. Kumar and B. Kumar, Some Fundamental Properties of Semigroups and their Classi�cations, Internal Journal of Mathematics Trends and Technology, Vol. 70 Issue 9, pp. 8�11, 2024. [15] R. Thakur, A Short Note on Completely Regular Semigroup, International Journal of Creative Research Thoughts , vol. 11, Issue 5, 2023. [16] R. J. Koch and A. D. Wallace, Stability in semigroups, Duke Math. J., vol. 24, pp. 193�196, 1957. [17] R. U. Ndubisi, and O. G. Udoaka, On left Restriction Semigroups, International Journal of Algebra and Statistics, vol. 5No. 1, pp. 59�66, 2016. DOI: 10.20454/ijas. 1083. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 8 [18] O. G. Udoaka, Rank of some Semigroup, Int. Journal of Applied Science and Math- ematical Theory, vol. 9, No. 3, pp. 90�100, 2023. www.iiardjournals.org. D.O.I: 10.56201/ijasmt.v9.no3.2023.pg90.100 [19] O. G. Udoaka, O. Tom, and A. Musa, On Idempotent Elements in Quasi-Idempotent Generated Semigroup, International Journal for Research Trend and Innovation, vol. 8, No. 11, pp. 2456�3315, 2023. [20] W. D. Burgess and R. Raphael, On Conrad's partial order relation on semiprime rings and on semigroups, Semigroup Forum vol. 16, pp. 133�140, 1978. [21] X. Mary, On the Structure of Semigroups Whose Regular Elements are Completely Regular. hal-04223812. 2023. [22] On quasi-separative Semigroup.Y. I. Krasilnikova and B. V. Novikove, arXiv:math/0311414v1 [math.GR], 2003. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 9 | September 2025 | http://ijojournals.com/index.php/m/index 9 Introduction Preliminaries Separative semigroup Semilattice of a Semigroup Results Conclusion