Some Study on the Topological Structure on Semigroups
KLEIDA HAXHI, TEUTA MYFTIU, KOSTAQ HILA
Department of Mathematical Engineering,
Polytechnic University of Tirana,
Rruga “S. Delvina”, Tirana, 1001,
ALBANIA
Abstract: - Some studies related to the topological structure of semigroups are provided. In, [3], considering
and investigating the properties of the collection of all the proper uniformly strongly prime ideals of a -
semigroup , such study starts by constructing a topology on using a closure operator defined in terms of
the intersection and inclusion relation among these ideals of -semigroup , which is a generalization of the
semigroup. In this paper, we introduce three other classes of ideals in semigroups called maximal ideals, prime
ideals and strongly irreducible ideals, respectively. Investigating properties of the collection , and of all
proper maximal ideals, prime ideals and strongly irreducible ideals, respectively, of a semigroup , we
construct the respective topologies on them. The respective obtained topological spaces are called the structure
spaces of the semigroup . We study several principal topological axioms and properties in those structure
spaces of semigroup such as separation axioms, compactness and connectedness, etc.
Key-Words: - Semigroup, prime ideal, maximal ideal, strongly irreducible ideal, structure space, hull-kernel
topology.
Received: October 26, 2021. Revised: September 21, 2022. Accepted: October 27, 2022. Published: November 16, 2022.
1 Introduction and Preliminaries
In [4], the notion of uniformly strongly prime ideals
in -semigroups is introduced. Chattopadhyay and
Kar, [3], considering and investigating the
properties of the collection of all proper
uniformly strongly prime ideals of a -semigroup ,
a topology on was constructed by means of a
closure operator defined in terms of intersection and
inclusion relation among these ideals of -
semigroup . The topological space is often
called the structure space of the -semigroup .
Since semigroups are generalizations of
semigroups, all the results obtained in, [3], hold for
semigroups. This kind of topological space has been
studied in different algebraic structures, [1], [2], [4],
[5], [6], [7], [8], [9], [10], [11]. Several principal
topological axioms and properties in this structure
space, such as separation axioms, compactness, and
connectedness, were studied.
In this paper, we study three other classes of
ideals in semigroups called maximal ideal, prime
ideal and strongly irreducible ideal, respectively.
Properties of the collection , and of all
proper maximal ideals, prime ideals and strongly
irreducible ideals respectively of a semigroup are
investigated. We construct the respective topologies
on them using a closure operator defined in terms of
the intersection and inclusion relation among these
ideals of the semigroup . Some principal
topological axioms and properties in those structure
spaces of semigroup are investigated.
Recall first the basic terms and definitions.
Let we consider the semigroup .
A nonempty subset of a semigroup is called
a sub-semigroup of if .
A nonempty subset of a semigroup is called
a right (left) ideal of if for all and ,
.
A nonempty subset of is called an ideal (or
two-sided ideal) if it is both a left ideal and a right
ideal.
An element in a semigroup is called identity
if .
An element in a semigroup is called zero
element if .
An element in a semigroup is called
idempotent element if .
The set of all idempotents of the semigroup is
denoted by .
A proper ideal of a semigroup is called a
prime ideal of if implies or
for any two ideals of .
An ideal of a semigroup is said to be full if
.
An ideal of a semigroup is said to be a prime
full ideal if it is both prime and full.
In, [3], the following theorem is proved.
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Theorem 1.1 [3, Theorem 2.8] Let be a
semigroup and be an ideal of . Then the
following statements are equivalent:
1. If and are ideals of such that
, then either or .
2. If are principal ideals of
such that , then either or
.
3. If , then either or
.
4. If and are two right ideals of such
that , then either or .
5. If and are two left ideals of such
that , then either or .
Definition 1.2 [3], [4] An ideal of a semigroup
is called a uniformly strongly prime ideal (usp
ideal) if contains a finite subset such that for all
, implies that or .
Theorem 1.3 [3], [4] Let be a semigroup.
Then every uniformly strongly prime ideal is a
prime ideal.
Throughout this paper will always denote
a semigroup with zero.
2 On Structure Space of Uniformly
Strongly Prime Ideals of Semigroup
In this section, let us recall some of the results and
definitions obtained in [3]. The philosophy of them
and their proofs will be useful for the results
obtained in this paper.
We denote by the collection of all uniformly
strongly prime ideals of a semigroup . For any
subset of (that is, subcollection), we define
.
It can be easily seen that .
Theorem 2.1 Let be any two subsets of .
Then
1. .
2. .
3. .
4. .
Definition 2.2 The closure operator gives
a topology on . This topology is called the
hull-kernel topology and the topological space
) is called the structure space of the
semigroup .
Remark 2.3 Let be a collection of prime
ideals of a semigroup . Then is either empty
or it is an ideal of but it need not be a prime ideal
of , in general. The following example shows it.
Example 2.4, [13], Let we consider the
semigroup , where . The
sets ( is prime) are
obviously prime ideals of . It is clear that the set
.
In [12] it is proved that the intersection of prime
ideals of a semigroup if it is not empty, is a
semiprime ideal of .
We have the following proposition:
Proposition 2.5 Let be a semigroup and
be a collection of prime ideals of such that
forms a chain. Then is a prime ideal of .
Definition 2.6 Let be a semigroup. The
structure space of is called irreducible if
for any decomposition , where
and are proper closed subsets of (whether
disjoint or non-disjoint), we have either or
.
Theorem 2.7 Let be a semigroup and be a
closed subset of . Then is irreducible if and only
if is a prime ideal of .
We denote by the collection of all uniformly
strongly prime full prime ideals of a semigroup .
We find that is a subset of and hence is
a structure space, where is the subspace topology.
In general, is not compact and
connected. But in particular, for the structure space
, we have the following theorem:
Theorem 2.8 Let be a semigroup. is a
compact space.
Theorem 2.9 Let be a semigroup. is a
connected space.
3 On Structure Space of Maximal
Ideals of Semigroup
In this section, the structure space of all maximal
ideals of a semigroup with identity 1 is
considered and studied.
A proper ideal of is maximal in if for any
ideal of with , then .
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Example 3.1 Let be a prime number. Let
. Then is a semigroup. Let is a
maximal ideal of .
Example 3.2 The set is a
maximal ideal of the semigroup .
Example 3.3 [13], Let
be the set of all residue classes mod 6. Then is
a commutative semigroup with identity, where
and (mod 6). We consider
. Then is a unique maximal ideal
of .
Let be the set of all maximal ideals in a
semigroup . We define two topologies on . For
every , we denote by the set of all maximal
ideals that contain , by the set , that is,
the set of all maximal ideals not containing . Let
be an ideal of , we denote by the set of all
maximal ideals containing .
We choose the family as a subbase
for open sets of . We shall refer to the resulting
topology on as -topology (in symbol, ).
Similarly, we take the family as a
subbase for open sets of (in symbol ).
Let and be two ideals of the semigroup .
We denote by the set of finite products of
members of .
Let be two distinct elements of . Then
we have . Therefore there are
such that and , so we have
and . Hence, we
have
Theorem 3.4 The structure space
is a -
space.
Let now be an element of , and
, then there is an element such that and
. Therefore, and .
This implies . Hence we obtain the
following
Theorem 3.5 The structure space is a
-
space.
Let be an ideal of and a generator of ,
then we have .
Therefore, the closed sets for the structure space
have the form , where are
ideals of .
Let
, if for some , then
and . This implies and we have
. Suppose that there is a maximal
ideal such that
, then
and
. Therefore, and do not
contain all . Therefore, since is a
maximal ideal, there are elements and
such that
.
Thus, we have
and . This implies .
Hence, by , we have , which is a
contradiction. This shows the following relation:
and we have the following
Theorem 3.6 The closed sets for are
expressed by sets , where is an ideal of .
By Theorem 3.5, we prove the following
theorem.
Theorem 3.7 The space is a compact
-
space.
Proof. Let be a family of closed sets in
with the finite intersection property, where
are ideals in . Therefore, any finite family of
does not contain the semigroup . Hence the ideal
generated by does not contain the identity 1 of
. This shows that is contained in a maximal ideal
. Hence . Therefore, since is
non-empty, is a compact space.
4 On Structure Space of Prime Ideals
of Semigroup
In this section, the structure space of all prime
ideals of a semigroup with identity 1 is
considered and the relation of and the structure
space of all maximal ideals of is investigated.
Throughout the section, we shall treat a
commutative semigroup with identity 1. An ideal
of is prime if and only if implies
or . Since has an identity 1, then any
maximal ideal is prime, therefore . We
notice here that a maximal ideal in a commutative
semigroup without identity may not be prime.
Example 4.1 The ideal of Example 3.3 is a
maximal ideal of and it is a prime ideal of .
The ideals of Examples 3.1 and 3.2
respectively, are maximal ideals but not a prime
ideal of .
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To introduce a topology in , we take
for every as an open base
of . We have the following.
Theorem 4.2 Let be a subset of , then
,
where is the closure of by the topology .
Proof. Let and let
be a neighbourhood of , then , and we have
. Therefore, there is a prime ideal
such that does not contain and . This
shows that . Thus we have proved that the
contains .
If a prime ideal is not in
, then . Hence, for
, we have and .
This shows and . Therefore
and hence . The proof is
complete.
A similar argument for relative to the -
topology implies the following.
Proposition 4.3 Let be a subset of , then
,
where is the closure of by the topology .
In a similar way to the proof of the Theorem 2.1,
we can prove the following
Theorem 4.4 The closure operation of
satisfies the following relations:
1. .
2. .
3. .
Proof. We shall prove only the last relation (3). By
Theorem 4.2, implies and hence
. Let , then and .
Hence and .
The sets and are ideals. If , for
any elements such that ,
we have and since is a prime ideal,
or , which is a contradiction. Therefore,
. Hence .
Theorem 4.5 The structure space is a -
space.
Proof. To prove that the structure space is a -
space, it is sufficient to verify the following
conditions:
1. .
2. .
3.
4.
implies
.
By the above theorem, it is sufficient to prove
that
implies
. By
,
then
. Similarly
and we have
.
Theorem 4.6 The structure space is a compact
-space.
Proof. Let be a family of closed sets such that
, then we have , where
. Indeed: Let us suppose that .
Then there is a maximal ideal containing .
Therefore for every . Hence for
every , and we have , which is a
contradiction. By , we have
. Hence
. If
, then for a prime ideal of
, we have and
hence
. Therefore we have
.
By the -radical of the semigroup , we
mean the intersection of all prime ideals of , that
is, . By the -radical of , we mean
the intersection of all maximal ideals of , that is,
.
From , we have . In the
following proposition we give a condition to be
.
Theorem 4.7 The subset of is dense in , if
and only if, .
Proof. Let for the topology . Then we have
.
Hence
.
Since , therefore we have
.
On the contrary, if , then and
. Therefore, there is a neighborhood of
such that . Hence is a
proper subset of . Therefore,
, which completes the proof.
Definition 4.8 If is the zero ideal (0), then
is said to be .
From the Theorem 4.7, we have the following
Theorem 4.9 If is -semisimple, is dense
in .
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5 On Structure Space of Strongly
Irreducible Ideals of Semigroup
In this section, the structure space of all strongly
irreducible ideals of a commutative semihypergoup
with identity 1 is investigated.
An ideal of a semigroup is called irreducible,
if and only if for two ideals implies
or . An ideal of a semigroup is
called strongly irreducible, if and only if
for every two ideals implies or .
From for any two ideals , it
follows that any prime ideals are strongly
irreducible and any strongly irreducible ideals are
irreducible.
Let be the set of all strongly irreducible ideals
of . From the above, it is clear that .
To give a topology on , we shall take
for every as an open base of .
Theorem 5.1 Let be a subset of , then we
have
where is the closure of by .
Proof. Let and let
. Let be an open base of , then, by the
definition of the topology , . Hence, we have
. It follows from this that there is a
strongly irreducible ideal of such that is not
contained in . Hence . Therefore and
.
To prove that , take a strongly irreducible
ideal such that . Then .
For an element , we have
and . Hence and for
all of . Therefore, and then we have
. Hence . The proof of the theorem is
complete.
We shall prove that the structure space for the
topology is a compact -space. To prove that is
a -space, it is sufficient to verify the following
conditions:
1. .
2. .
3.
4. implies .
Conditions (1) and (2) are clear and implies
. From this relation we have .
For some element of of , suppose that
and . From Theorem 5.1, we have
and .
and are ideals. If , by the
definition of , or . Hence
. This shows .
To prove that implies , we shall
use condition (1). Then and by the
definition of closure operation, we have .
Similarly, we have and . Therefore,
we complete the proof that is a -space.
We shall prove that is a compact space. Let
be a family of closed sets with empty intersection.
Let , suppose that
, then
there is a maximal ideal containing the ideal
. Therefore, we have for every . By
the definition of for every . Hence
, which contradicts our hypothesis of .
Therefore,
. Therefore, we have
. Therefore, we
have . If
,
for all strongly irreducible ideals of
,
and
. If
, we can easily prove that is a compact space. If
contains a proper strongly irreducible
ideal , we have
, which is a
contradiction to
. Therefore
. Hence is a compact space. Thus,
we have proved the following.
Theorem 5.2 The structure space is
compact -space.
By of a semigroup, we mean
the intersection of all strongly irreducible ideals of
it, that is, . From , we have
.
Theorem 5.3 The subset of is dense in , if
and only if .
Proof. Let for the topology , then we have
.
Hence, we have
.
On the other hand, . This shows
.
Conversely, suppose that , then there is
a strongly irreducible ideal such that and
. Therefore, there is a neighborhood of
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that does not meet . Therefore, is a
proper subset of , and we have
.
Corollary 5.4 The subset of is dense in ,
if and only if .
Corollary 5.5 Let be a semigroup with 0. If
is -semisimple, then and are dense in .
6 Conclusions
In this paper, we investigated three other classes of
ideals in semigroups called maximal ideals, prime
ideals and strongly irreducible ideals, respectively.
Properties of the collection , and of all
proper maximal ideals, prime ideals and strongly
irreducible ideals respectively of a semigroup
were investigated. We constructed the respective
topologies on them using a closure operator defined
in terms of intersection and inclusion relation
among these ideals of the semigroup . Some
principal topological axioms and properties in those
structure spaces of semigroup were investigated.
In future work, one can develop and extend the
study of these structure spaces in -semigroups or
further in semihypergroups, -semihypergroups and
other kinds of hyperstructures.
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