




































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 

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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.

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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.

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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:

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� 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.

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� 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:

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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.

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	Introduction
	Preliminaries
	Separative semigroup
	Semilattice of a Semigroup
	Results
	Conclusion

