Interdisplinary Journal of Research and Development
“Alexander Moisiu “ University, Durrës, Albania
Vol (IV), No.2, 2017
_____________________________________________________________________________________________ Paper presented in 1-st International Scientific Conference
on Professional Sciences, “Alexander Mosiu” University, Durres
November 2016
GREEN TYPE THEOREMS FOR LE-SEMIGROUPS SATISFYING CERTAIN
COMMUTATIVITY CONDITIONS
AIDA SHASIVARI1, ELTON PASKU2
1
Polytechnic University of Tirana, Faculty of Mathematical Engineering, Tirana, Albania
2
University of Tirana, Faculty of Natural Sciences, Tirana, Albania
Corresponding author e-mail : [email protected]
Abstract
The famous Green's theorem in a plain semigroup S states that if an ℋ −class H satisfies the
condition H ∩ 𝐻 ! ≠ 0 (commonly known as the Green condition), then H is a subgroup of S.
Any attempt to generalize Green's theorem to le-semigroups or other types of ordered
semigroups is useless since it has been proved from several authors that there are examples of
le-semigroups containing ℋ classes satisfying the Green condition which do not even form
subsemigroups.
The aim of this paper is to prove Green type theorems for le-semigroups satisfying certain mild
commutativity conditions. More specifically, we prove that if S is a le-semigroup in which
quasi-ideal elements commute, then any ℒ −class (resp. ℋ-class) satisfying the Green
condition is a subsemigroup of S. To achieve these results we consider the local Green relations
defined in principal ordered ideals generated by left ideal elements and show that the usual
classes which satisfy the Green condition include in certain local classes which turn out to be
subsemigroups.
Key words: Ordered semigroups, Green's relation, quasi-ideal elements, ordered ideals.
1. Introduction and preliminaries
The study of Green's relations in the theory of
semigroups has been very influential since they
were defined in mid 50's. Green's relations
become useful when we want to see how far is a
given semigroup from simpler structures such as
simple or completely simple semigroups,
semilattices of groups, regular or inverse
semigroups and so on.
One of the most remarkable results related to
these relations is the Green's theorem which
states that any ℋ-class Η of a semigroup S that
satisfies the condition 𝛨 ∩ 𝛨 ! ≠ ∅ (commonly
known as the Green condition) forms a subgroup
of S. It is obvious that semigroups that satisfy
Green's condition for every ℋ-class are locally
like groups. They are known in fact as
completely regular semigroups and play an
essential role in the theory of semigroups.
There is a parallel theory to that of ordinary
semigroups mostly developed from Kehayopulu
and her descendants which is called the theory
of
ordered semigroups. An ordered semigroup or a
po-semigroup is a triple (S, ≤ ,∙) where ≤ is an
order relation in S and ∙ is an associative
multiplication in S that satisfy the properties
a≤b ⇒xa≤ xb and ax ≤ bx for all 𝓍 𝜖S:
If (S, ≤ , ∙) is a po-semigroup possessing a
greatest element ℯ, then it will be called a
𝑝𝑜𝑒-semigroup and it is called a 𝑉𝑒-semigroup
if (S, ≤ ) is an upper semilattice. If in addition it
happens that (S; ≤) is a lattice (the meet ⋀ and
the joint ∨of any two elements of S exists), then
1 Green type theorems for……….
A. Shasivari & E.Pasku
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1 2 3
∉ 𝐻, whence H fails to be a
1 1 1
subsemigroup.
In these circumstances it is appreciated to find
out moderate conditions under which ℋ-classes
or more generally ℒ (resp. ℛ)-classes form
subsemigroups. In the present paper we propose
the condition: any two left ideal element
commute with each other. This condition is not
unnatural. In [4] Kehayopulu and Tsingelis
consider ordered semigroups having the property
that any two left ideals 𝐿! and 𝐿! satisfy
(𝐿! 𝐿! ]=𝐿! ∩ 𝐿! , which clearly implies the
analogue of our condition, and characterize them
as those ordered semigroups that decompose as a
semilattice
of
left-simple
semigroups,
generalizing thus a result of Saitô for
semigroups without order in [6]. Also in [1]
Kehayopulu has shown that in le-semigroups our
proposed condition together with left regularity
is equivalent to 𝑙 ⊳ 𝑟 which in turn means the
left regularity and that the left ideal elements are
at the same time right ideal elements.
We prove in our theorem 2.2 that any ℒ-class
that satisfy the Green condition forms a
subsemigroup provided that any two left-ideals
commute. Regarding the analogue of Green's
theorem for ℋ-classes in le-semigroups we
propose another condition: any two quasi-ideal
elements commute with each other.
Also this condition is not unnatural. It obviously
generalizes our previous condition as left ideal
elements are quasi-ideal elements. Also it is
known that in ordinary semigroups the property
𝑄! 𝑄! = 𝑄! ∩ 𝑄! for every quasi ideals 𝑄! 𝑄!
characterizes Clifford semigroups.
S will be called a le-semigroup. The standard
notation for the le-semigroup in this case is
(S, ∙ ,∧ ,∨) . Here the order relation is not made
explicit but it is understood that
𝑏!=
a ≤b iff a ∧b = a.
An element x of a poe-semigroupS is regular if
x≤xex.
Kehayopulu [2], [3] has defined the following
equivalences in a given le-semigroup 𝑆,∙ ,∧ ,∨ 𝓛= 𝓍; y ∈ 𝑆 𝑥 𝑆 𝓍 ∨ 𝑒𝑥 = 𝑦 ∨ 𝑒𝑦
𝓡= 𝓍; y ∈ 𝑆 𝑥 𝑆 𝑥 ∨ 𝓍𝑒 = 𝑦 ∨ 𝑦𝑒
𝓗= 𝓡 ∩ 𝓛
These relations were latter called from Petro and
Pasku in [5] the Green-Kehayopulu relations and
generalize the usual Green's relations in ordinary
semigroups. They have been proved to be very
helpful in generalizing many interesting results
from the theory of ordinary semigroups to that
of ordered semigroups but fail a key property
their analogue in plain semigroups has, namely
the property depicted in the Green's theorem. In
fact ℋ-classes that satisfy the green condition
need not be subsemigroups. The following
example which was proposed to the authors of
[5] by M.V.Volkov shows the existence of an
ℋ-class in the le-semigroup of all orderpreserving mappings of a finite chain that
satisfies the Green condition which fails to be a
sub semigroup.
Example 1.
Consider the le-semigroupS of all orderpreserving transformations of the finite chain
1 < 2 < ⋯ < 𝑛 . It is easy to describe the
Green relation ℋon 𝑆:
2. The main results
We give below some basic notions that will be
used throughout the next subsection.
Let S stands for an le-semigroup 𝑆,∙ ,∧ ,∨ . We
define the map
(a; b) 𝜖ℋ if and only if n.a = n.b
Thus, for 𝑛 = 3 the transformations
𝑎=
1
1
2
2
3
2
an 𝑏 =
1
1
2
1
ℓ𝓁: 𝑆 ⟶ 𝑆 by setting ℓ𝓁(𝓍) = 𝑒𝑥 ∨ 𝑥;
3
2
and show that ℓ𝓁(x) is the least left ideal element
which is bigger than 𝓍. That ℓ𝓁(𝓍) is an ideal
element follows from the fact that
are ℋ-equivalent. Their ℋ-class H satisfies
Greens condition since 𝒶 is an idempotent, but
2 Interdisplinary Journal of Research and Development
“Alexander Moisiu “ University, Durrës, Albania
Vol (IV), No.2, 2017
_____________________________________________________________________________________________ ℯ ∙ (e𝓍 ∨ 𝓍 ) = ℯ ! 𝓍 ∨ ℯ𝓍 = ℯ𝓍 ≤ ℓ𝓁(𝓍)
Proof. The if part is obvious so it remains to
prove that if b, c,bc∈L, then 𝜆 = 𝜆! . Since from
lemma 2.1, b, c≤ 𝜆, then bc ≤ 𝜆! . Passing to left
ideal element generated from the both sides in
the last inequality we get that ℓ𝓁(bc) ≤ ℓ𝓁(𝜆! ).
But 𝜆! is a left ideal element as well and so
ℓ𝓁(bc)≤ 𝜆! .
Recalling that bc∈ 𝐿! , we have 𝜆 ≤ 𝜆! which
together with the obvious inequality 𝜆! ≤ 𝜆
imply the desired 𝜆 = 𝜆! .
It is interesting to ask under what conditions a
ℒ-class satisfying the Green condition forms a
subsemigroup of S. The answer to the case when
the class is required to be a group is rather easy
and is proved in proposition 2.1.
Before we need the following.
Also the definition of ℓ𝓁(𝓍) shows that ℓ𝓁(𝓍) ≥ 𝓍.
It remains to show that if 𝜆 is a left ideal element
of S such that 𝓍 ≤ 𝜆, then ℓ𝓁(𝓍)≤ 𝜆. Indeed,
ℓ𝓁(𝓍) = ℯ𝓍 ∨ 𝓍
≤ ℯ𝜆 ∨ 𝜆 since 𝓍 ≤ 𝜆
≤ 𝜆 since𝜆
is a left ideal element.
Now we are ready to define the relation ℒ in S.
We say that two elements 𝑥, 𝑦 ∈ 𝑆 are 𝓛related, written as (𝑥, 𝑦) ∈ ℒ if and only if
ℓ𝓁(𝓍) = ℓ𝓁(𝑦). In other words
𝓛=
𝓍; y ∈ S x S ℓ𝓁(𝓍) = ℓ𝓁(𝑦)
It is straightforward that ℒ is an equivalence
relation, hence we can talk about its ℒ-classes.
A key property of the ℒ-classes is depicted in
the following.
Lemma 2.2. Assume that the ℒ-class L is a
subsemigroup of S. If 𝜆 is the representative left
ideal element of L, then for every element 𝓍 ∈ 𝐿,
𝜆𝓍 = 𝜆 = ℯ𝓍
Lemma 2.1. Every ℒ-class L of S has a unique
ideal element ℓ𝓁(𝒶) where 𝒶 is an arbitrary
element of L.
Proof. It is obvious that 𝜆𝓍 is a left ideal
element of S. Since L is assumed to form a
subsemigroup and 𝓍 ∈ L, then 𝜆𝓍 ∈ 𝐿 and so by
lemma 2.1 we get 𝜆𝓍 = 𝜆.
Further,
𝜆 = 𝜆𝓍 ≤ ℯ𝓍 ≤ 𝜆
proving that 𝜆𝓍 = ℯ𝓍.
The following is an analogue of proposition 2.3
of [5].
Proof. First, for any 𝒶 ∈L we show that ℓ𝓁 𝒶 ∈
ℐ. Indeed, since ℓ𝓁 𝒶 is a left ideal element,
ℓ𝓁(ℓ𝓁 𝒶 )= ℓ𝓁(𝒶) and then the definition of ℒ
shows that (𝒶; ℓ𝓁(𝒶)) ∈ ℒ or equivalently,
ℓ𝓁(𝒶) ∈ L𝒶 . Finally, if 𝜆 ∈ 𝐿 is a left ideal
element, then on the one hand ℓ𝓁(𝜆) = 𝜆 and on
the other
ℓ𝓁(𝜆) = ℓ𝓁 𝒶 since 𝒶 ∈L, and so we get that 𝜆=
ℓ𝓁(𝒶).
The element ℓ𝓁 𝒶 of the previous lemma is
called the representative left ideal element of L.
Proposition 2.1 .A ℒ-class L is a subgroup of S
if and only if L consists of a single idempotent
element.
Proof. The if part is obvious. Now assume that L
is a subgroup of S and let 𝜆 be the representative
left ideal element of L, 𝜆 its inverse and 𝑖 the
unit element. We see first that 𝑖 is a left ideal
element. Indeed, ℯ𝑖 = ℯ𝜆𝜆
2.1 𝓛-classes being subsemigroups
A ℒ-class L is said to satisfy the Green condition
if there are b,c∈L such that bc∈L.
The representative left-ideal element plays a key
role in regard with the Green condition as the
following shows.
= 𝜆𝜆 from lemma 2.2 = 𝑖
providing the claim. But 𝜆 is the only left ideal
element of L so 𝜆 = 𝑖. Further if 𝓍 ∈ 𝐿, then
𝑖 = 𝜆 from above
Theorem 2.1. A ℒ-class L satisfies the Green
condition if and only if its representative ideal
element 𝜆 is an idempotent.
= 𝜆𝓍 from lemma 2.2
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A. Shasivari & E.Pasku
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To see this we note first that for every
𝑥 ∈ 𝐿 = 𝐿! , 𝜀𝜆 = 𝜀𝓍.
= 𝑖𝓍 = 𝓍 since 𝑖 is the unit,
thus proving that L = 𝑖 is a singleton.
In what follows we will consider le-semigroups
that satisfy the condition: any two left ideal
elements commute with each other. We prove in
our theorem 2.2 that in such semigroups any
ℒ -class that satisfies the Green condition is a
subsemigroup. Assume we are given a lesemigroup 𝑆,∙ ,∧ ,∨ with the greatest element 𝜀
and let 𝕃 be the set of left ideal elements of S.
For every ℯ ∈ 𝕃 we let ]ℯ] = 𝓍 ∈ 𝑆: 𝑥 ≤ 𝑒 . It is
clear that ]e] is a subsemigroup of S since
if 𝑥𝑦 ∈]𝑒], then𝑥𝑦 ≤ 𝑒𝑒 ≤ 𝜀𝑒 ≤ 𝑒. With the
order relation of S restricted in ]𝑒], the latter
forms a le-semigroup with the greatest element
𝑒.
We denote by 𝓛(𝓮) the relation 𝓛 defined in ]ℯ].
With the above notations
we have the following.
It follows that for the ideal element of ] 𝜆]
generated from 𝓍 we have that,
Lemma 2.3. If ℯ ∈ 𝕃 is an idempotent, then 𝐿ℯ
is a subsemigroup.
Proof. First, for every 𝑥 ∈ 𝑆, 𝑒𝑥 is left ideal
element of S since 𝜀(𝑒𝑥) = (𝜀𝑒)𝑥 ≤ 𝑒𝑥.
(ℯ)
which proves that 𝜆 = 𝜀𝑎𝑏 ∨ 𝑎𝑏 or equivalently
that 𝑎𝑏 ∈ 𝐿! concluding the proof.■
Second, it is easy to see that for every 𝑥 ∈
(ℯ)
𝐿ℯ we have 𝑒 = 𝑒𝑥.
(ℯ)
Further, if 𝑥, 𝑦 ∈ 𝐿ℯ then we can write
Now we consider le-semigroups that satisfy the
condition any two quasi-ideal elements commute
with each other and prove that in such lesemigroups any ℋ-class that satisfies the Green
condition is a subsemigroup. Assume we are
given a le-semigroup 𝑆,∙ ,∧ ,∨ with the greatest
element ℯ and let Q be the set of ideal elements
of S. For every 𝜀 ∈ 𝑄 we let ]𝜀] = 𝓍 ∈ 𝑆: 𝑥 ≤ 𝜀 .
It is clear that ] 𝜀] is a subsemigroup of S since
if𝓍, 𝑦 ∈ ]𝜀], 𝓍𝑦 ≤ 𝜀𝜀 ≤ 𝜀𝑒 ≤ 𝜀.
With the order relation of S restricted in ]𝜀], the
latter forms a le-semigroup with the greatest
element 𝜀. We denote by ℋ (!) the relation
ℋdefined in ]𝜀].
𝜆𝓍 ∨ 𝓍= (𝜀𝜆)𝓍 ∨ 𝓍
= (𝜆𝜀)𝓍 ∨ 𝓍 left ideals commute
= 𝜆 𝜀𝓍 ∨ 𝓍 = 𝜆𝜆 ∨ 𝓍 from lemma 2.2
= 𝜆 since 𝓍 ≤ 𝜆
(!)
which shows that 𝓍 ∈ 𝐿! thus proving the
(!)
inclusion L = 𝐿! ⊆ 𝐿! .
(!)
(!)
Let 𝑎, 𝑏 ∈ 𝐿! . Since 𝐿! ⊆ 𝐿! and 𝐿! is a
subsemigroup from lemma 2.3 we have that
𝜆 = 𝜆𝑎𝑏 ∨ 𝑎𝑏. But
𝜆𝑎𝑏 ∨ 𝑎𝑏 ≤ 𝜀𝑎𝑏 ∨ 𝑎𝑏
≤ 𝜀𝜆! ∨ 𝜆! since 𝑎, 𝑏 ≤ 𝜆
= 𝜀𝜆 ∨ 𝜆 since 𝜆 = 𝜆! = 𝜆
2.2 H-classes being subsemigroups
𝑒 = 𝑒𝑥 𝑒𝑦
= 𝑒(𝑒𝑥)𝑦 left ideals commute
= 𝑒 𝑥𝑦 ≤ 𝑒 ! = 𝑒,
(ℯ)
from which follows easily that 𝓍𝑦 ∈ 𝐿ℯ .
Theorem 2.2. If S is a le-semigroup in which
any two left ideal elements commute, then any
ℒ-class satisfying the Green condition is a
subsemigroup of S.
Proof. Assume that L is a ℒ-class in S satisfying
the Green condition. Write 𝜆 for the
representative left ideal element of L which from
lemma 2.1 has to be idempotent and from lemma
2.2 satisfies 𝜆 = 𝜀𝜆. We claim that
Theorem 2.3. If S is a le-semigroup in which
quasi-ideal elements commute, then any ℋ-class
that satisfies the Green condition is a
subsemigroup of S.
Proof. Let 𝐻! be the given ℋ-class where 𝑞 is
its representative quasi-ideal. From [5] we know
(!)
L = 𝐿! ⊆ 𝐿! .
4 Interdisplinary Journal of Research and Development
“Alexander Moisiu “ University, Durrës, Albania
Vol (IV), No.2, 2017
_____________________________________________________________________________________________ (ℯ!ℯ)
that 𝑞 ≤ ℯ𝑞 and 𝑞 ≤ 𝑞ℯ. It follows that 𝑞 ≤ ℯ𝑞ℯ,
hence 𝑞 ∈]ℯ𝑞ℯ].
(ℯ!ℯ)
Let 𝐻! the ℋ-class of 𝑞 in the lesemigroup ]ℯ𝑞ℯ]. We claim first that 𝐻! ⊆
previous equality show that 𝓍𝑦 ∈ 𝐻! proving
the second claim. Finally we prove that 𝐻! is a
(ℯ!ℯ)
subsemigroup. Let 𝓍, 𝑦 ∈ 𝐻! . Since 𝐻! ⊆ 𝐻!
and the latter is a subsemigroup, then 𝓍𝑦 ∈
(ℯ!ℯ)
𝐻! , therefore we have
𝓍𝑦 ℯ𝑞ℯ =
𝑞 ℯ𝑞ℯ = 𝑞ℯ.
We can now write 𝑞ℯ = 𝓍𝑦 ℯ𝑞ℯ ≤ 𝓍𝑦 ℯ ≤
𝑞 ! ℯ ≤ 𝑞ℯ which proves that (𝓍𝑦, 𝑞) ∈ ℛ.
Similarly (𝓍𝑦, 𝑞) ∈ ℒ hence 𝓍𝑦 ∈ 𝐻! proving
the theorem.■
(ℯ!ℯ)
𝐻! .
Indeed, if 𝓍 ∈ 𝐻! , then from [5] we have
𝓍ℯ = 𝑞ℯ and ℯ𝓍 = ℯ𝑞. Using the first equality
we see that
𝓍 ℯ𝑞ℯ ∨ 𝓍 = (𝓍ℯ) 𝑞ℯ ∨ 𝓍 = 𝑞ℯ 𝑞ℯ ∨ 𝓍 =
𝑞ℯ ∨ 𝓍 = 𝑞ℯ = 𝑞 ℯ𝑞ℯ ∨ 𝑞
(1)
References
[1] Kehayopulu, N. (1973). A characterization of
semiregular and regular m-lattices. In:
Panayotopoulos, A. editor. C. Caratheodory
Inter. Sym. The Greek Math. Soc., pp. 282-295.
[2]Kehayopulu,N.(1989).
On minimal quasi ideal elements in poe −
semigroups, Mathematica Japonica 34, 767774.
[3]Kehayopulu,N.(1990).
On Filters generated in poe − semigroups,
Mathematica Japonica 35, 789-796.
[4] Kehayopulu N, Tsingelis M. (2013). On
Ordered Semigroups which are Semilattices of
Left Simple Semigroups. Mathematica Slovaca;
63(3): 411-416
[5] Petro P, Pasku E. (2002). The GreenKehayopulu relation 𝓗in le −semigroups.
Semigroup Forum; 65: 33-42.
[6] Saitô T. (1973). On semigroups which are
semilattices of left simple semigroups,
Math.Japonica; 18: 95-97.
In a symmetric way using the second given
equality above we find that
ℯ𝑞ℯ 𝓍 ∨ 𝓍 = ℯ𝑞 (ℯ𝓍) ∨ 𝓍 = ℯ𝑞 ℯ𝑞 ∨ 𝓍 =
ℯ𝑞 ∨ 𝓍 = ℯ𝑞 = ℯ𝑞ℯ 𝑞 ∨ 𝑞
(2)
(ℯ!ℯ)
Now, (1) and (2) imply that 𝓍 ∈ 𝐻!
our first claim.
(ℯ!ℯ)
Second we claim that 𝐻!
proving
forms
subsemigroup of S. We remark that
a
(ℯ!ℯ)
𝐻!
(ℯ!ℯ)
𝐻!
satisfies the Green condition since
contains 𝐻! and the latter satisfies the condition.
(ℯ!ℯ)
Let now, 𝑦 ∈ 𝐻! . Since(ℯ𝑞ℯ)𝓍 = ℯ𝑞ℯ 𝑞 =
ℯ𝑞 then
(ℯ𝑞ℯ)𝓍𝑦 = ℯ𝑞 𝑦 .
Further we have ℯ𝑞 ≥ ℯ ! 𝑞 = ℯ𝑞ℯ
commute
quasi-ideals
= ℯ𝑞 ℯ𝑞ℯ ≥ ℯ𝑞 𝑦 = (ℯ𝑞ℯ)(𝓍𝑦)
= (ℯ𝓍) ℯ𝑞 𝑦 quasi-ideals commute
≥ ℯ ℯ𝓍 ℯ𝑞 𝑦 ≥ ℯ𝓍 ℯ𝑞ℯ 𝑦 as before
=
ℯ𝓍 ℯ𝑞
since (ℯ𝑞ℯ)𝑦 = ℯ𝑞ℯ 𝑞 = ℯ𝑞
≥ ℯ ℯ𝓍 ℯ𝑞 = ℯ𝓍 ℯ𝑞ℯ
quasi-ideals
commute = ℯ𝑞ℯ
since 𝓍 ℯ𝑞ℯ = 𝑞ℯ ≥ ℯ𝑞
which proves that (ℯ𝑞ℯ)(𝓍𝑦) = ℯ𝑞.
Now we prove that (𝓍𝑦)(ℯ𝑞ℯ) = 𝑞ℯ.
𝑞ℯ ≥ 𝑞ℯ ! = ℯ𝑞ℯ quasi-ideals commute
=
ℯ𝑞ℯ 𝑞ℯ ≥ 𝓍 𝑞ℯ
=
𝓍(𝑦 ℯ𝑞ℯ ) ≥
𝓍 𝑞ℯ 𝑦ℯ
quasi-ideals
commute
≥ 𝓍 𝑞ℯ 𝑦ℯ ℯ = 𝓍 ℯ𝑞ℯ 𝑦ℯ
quasi-ideals
commute = 𝓍 ℯ𝑞ℯ ℯ = ℯ𝓍ℯ𝑞ℯ quasi-ideals
commute = ℯ𝑞ℯ ≥ 𝑞ℯ which shows that
(𝓍𝑦)(ℯ𝑞ℯ) = 𝑞ℯ. This together with the
5 Green type theorems for……….
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