Idealization of a Semigroup by Using Bruck-Reilly Monoids

: The aim of this paper is to create a new semigroup by defining a special idealization operation on semigroups. Additionally, by considering amalgamation, we will present novel distinguishing results on idealization over semigroups. Initially, we will combine the definitions of idealization and the Bruck-Reilly extension. Consequently, by integrating these two structures, we will provide a significant result on idealization in semigroups that will be crucial for future studies. Finally, we will conclude by discussing the ideal extension for the new semigroup structure, namely idealization.


Introduction and preliminaries
One of the notable advantages of considering a new and more general construction is the unification of existing results under a novel structure.Moreover, it offers a concise specification, facilitates the derivation of new designs, and provides an economical approach to proving the correctness of certain properties.Additionally, when we view extensions as a means of combining known structures to create a new structure, this approach yields similar benefits in a distinct and effective manner.As an illustrative example, in [1][2][3], the authors introduced a new semigroup called  and characterized it by extensively studying its properties.
One of the most important constructions, as indicated in the above paragraph, is idealization (trivial extension or ringification), which is built on a ring with two operations and has been used to produce some interesting results.For a commutative ring R with the identity and an R-module M, the idealization R(+)M of M was first introduced by Nagata in [4] for all 1 2 , r r R ∈ and 1 2 , .m m M ∈ Idealization is the process of enclosing M in a commutative ring A so that its structure as a R-module is almost identical to its structure as an A-module, or as an ideal of A. Hence, it is helpful for generalizing results from rings to modules, reducing results about submodules to the ideal case, and creating examples of commutative rings with zero divisors.Although there are so many important studies about idealization in the literature (see, for instance, in [5][6][7][8]), different versions of this subject are still of interest.For instance, for a ring R, an ideal I, and an R-module M, the authors (in [9]) defined the so-called amalgamated duplication which produces a new ring as well as satisfying many properties coinciding with the idealization if M = I.Meanwhile, in [10], it has been proposed that another possible generalization of the duplication is as follows: Let R and U be commutative rings with unity; let J be an ideal of U; and let f: R → U be a ring homomorphism.In this setting, there exists a subring which is called the amalgamation of R with U along J with respect to f.This construction is a generalization of the amalgamated duplication, and so it contains the idealization as well.
In addition to the above studies, a survey was presented in [11] on some ring constructions and showed how one might produce some analogous semigroup constructions.In light of the idea, the main purpose of this paper is to introduce another approach to construct the idealization for semigroups by using the (single) semigroup operation.However, the biggest challenge when doing this is reducing binary operations on rings to a single operation on semigroups.To solve it, we will use the operation defined on Bruck-Reilly extensions in our construction (see Theorem

below).
Hence, let us remind ourselves of the Bruck-Reilly monoid that will definitely be needed in this paper.Assume that A is a monoid and θ is an endomorphism such that Aθ is in the  -class (this class is a binary relation defined on the elements of a semigroup, where two elements are related if and only if they generate the same principal right ideals.See [12] Chapter 2 of the identity 1 A of A. Thus, for the set of all non-negative integers 0 ,  the set 0 , where max( , ) t n m′ = and θ 0 is the identity map on A, forms a monoid with the identity (0,1 , 0).

A
Then this monoid is called the Bruck-Reilly extension of A, determined by θ and denoted by ( , ) BR A θ (cf. [13-15]).The Bruck-Reilly extension is considered one of the fundamental constructions depending on the isomorphism and is presented as a characterization for the theory of semigroups.For example, between any bisimple regular w-semigroup and the Bruck-Reilly extension of a group or between any simple regular w-semigroup and the Bruck-Reilly extension of a finite chain of groups, there exist isomorphisms (see [14,16]).We may suggest [15,17,18] for some other examples of characterizations via Bruck-Reilly extensions to the reader.

A new semigroup via Bruck-Reilly operation
As stated in the previous section, our goal is to use the Bruck-Reilly operation to build a new structure based on the idealization of a semigroup.To do that, we will combine the operations presented in (1) and ( 2), and hence we will capture the infrastructure relationship between semigroups and (sub-Bruck-Reilly) monoids similarly as in the idealization between modules and rings.
For any two submonoids SM 1 and SM 2 of the monoid M, let us assume that the Bruck-Reilly extension ( , ) such that, for arbitrary elements, 3) denotes the Bruck-Reilly operation given in (2).With this approach, a connection will be established between idealization and this new semigroup since the operation given in (2) and used in (3) has the same meaning as (1) in the definition of idealization.This approach certainly fits with the way defined in [11] to show how it may produce some analogies on a semigroup construction.
On the other hand, when a monoid is trivial, its Bruck-Reilly extension is isomorphic to the bicyclic monoid.This is very valuable in that the semigroup formed by the trivial monoid belongs to an important type of semigroup.
Thus, we have the following first pre-result of this paper: Proposition 2.1.The set i SB B × with the operation given in (3) defines a semigroup which will be denoted by ( ) .i SB B  Proof.It suffices to prove the well-defined and associative properties of (3).Suppose that Now, by taking into account (2), let us calculate a  c, a  d and c  b separately.Firstly, we have where and θ is a monoid homomorphism (no necessary to know the rule) having one of the possibilities Actually, one of these homomorphisms must exist in (4) due to the definition.Secondly, (which certainly exists), but since it provides the trivial case, the other option has been chosen (see Figure 1).
and χ must be a homomorphism either in the form by the same idea as in the previous paragraph, we certainly have a homomorphism Here, the existence of the homomorphism χ from SM 2 to M is obvious (see Figure 1).Now, the next step is to combine the above elements under the Bruck-Reilly operation  as presented in (3) (and so by (2)).Thus, which is equal to the element  is a monoid with the identity ( ) (0,1 ,0),(0,1 ,0) .

M M
Proof.By the assumption, let .

M
if we apply the operation in (3), then we obtain that 0 d n   is actually equal to 0 ((0,1 , 0) (0,1 , 0)), ((0,1 , 0) (0,1 , 0)) ((0,1 , 0) ( , , )) (0,1 , 0), ((0,1 , 0) (0 0 , Recall that an idempotent element of a semigroup S is defined as a 2 = a for any a ∈ S, and if every a ∈ S is an idempotent element, S is called the idempotent semigroup or band.Recall that the notation E(S) denotes the set of idempotents.The idempotent elements and bands play an important role in semigroup theory.Since our new semigroup ( ) i SB B  is not a band (see Lemma 2.3 below), we give our attention to showing that the set ( ( ) is a band (and a semilattice) under the condition that M is a commutative monoid (see Theorem 2.5 below).
The following lemma gives an explicit description of the idempotent elements of ( )   Similarly, for Case I, we have the following three subcases for Case II: (i)′ For the first components, we have x -z -f + d + t'' and this implies the equality x = z = d = f.(ii)′ For the second components, we have  The only option to achieve the final equality is y ∈ SM 2 = 1 M , which clearly implies e ∈ E(M).(iii)′ For the third components, we have f -d -x + z + t′′ = f, which gives the equality x = z = d = f as in (i)′.The sufficiency part: Conversely, let us assume that conditions 2 , y SM x z d f ∈ = = = and e ∈ E(M) are all held.Thus, we observe all elements of ( ) By a simple calculation, we clearly obtain , s s s =  as required.By Lemma 2.3, we basically say that the set of idempotent elements of the semigroup {( In the same way as in the proof of Proposition 2.1, we can show that ( ( ) is a semigroup under the operation defined in (3).Furthermore, as a consequence of Theorem 2.2, the elements in the set (6) 6) defines a semigroup.Moreover, it defines a monoid if we choose x = 0. Nevertheless, since a commutative idempotent semigroup is called a semilattice (cf.[12]), we then have the following theorem as a next step to Proposition 2.4.
Since M is commutative, the expressions in ( 7) and ( 8) are equal to each other, which yields ( ( ) is a commutative semigroup and so a semilattice, as required.

An ideal extension for ( )
i

SB B 
In this section, we will focus on a special extension of the semigroup ( ) defined in Proposition 2.1.Extensions were first systematically studied by Clifford [19], who gave the first general structure theorem in the case when S is weakly reductive (see [20] Theorem 4.21).Later on, this result was extended to arbitrary semigroups by Yoshida [21].
Let K and T be disjoint semigroups such that K has an identity and T has a zero element.A semigroup  is called an (ideal) extension of K by T if it contains K as an ideal and if / .K T ≅  Special types of ideal extensions, namely strict and pure, have also been introduced in the literature.We may refer, for instance, to [22,23]  [ ] i SB B   In fact, the coming theorem will use this monoid and will be a fundamental structure to obtain an ideal extension for our new semigroup.
Before presenting the following result, it should be noted that the index sets I and J in these Rees matrix semigroups are considered the non-negative integer set 0  in this theorem.Theorem 3.1.Suppose that RM 1 and R are two Rees matrix semigroups, which are defined on the sets  respectively.Then, we have ( ) For arbitrary elements, the equality (0,1 , 0), ( β must be a homomorphism with the form either 1 ′ ∉ Thus, by the fact that y 2 is an element of both M and SM 1 , we obtain the required homomorphism as

1 Figure 1 .
Figure 1.While the scanned area represents M \SM 1 , bold characters represent the monoid and its submonoids so as not to cause confusion Replacing t by d, the last equality is equal to For Case I, there further exist the following three subcases: (i) For the first component, we already have x -z + t = x, which implies the equality t = x = z.(ii) For the second component, we have ( ) ( ) .

=
But, by (i), since t = x = z, and since θ 0 is the identity homomorphism, For the third component, we have z − x + t = z, which yields x = z, as in (i).
)′, since x = z = d = f, we have t′ = z = d, the map f is well defined.By the assumption on the element a, we have by the associativity of ,  we obtain f is a homomorphism.Furthermore, since for all ( ) Im , f =  which implies that f is an epimorphism.Finally, let us consider the monoid ( ) is an element of SM 1 or SM 2 = 1 M .On the other hand, for the element (a  d)  (c  b) in (5), it can be observed that the element .
for a detailed introduction to ideal extensions and examples illustrating the strict or pure types.Let us suppose that  by  .Also, for a fixed a ∈ SB 1 ⊂ B, suppose that the element a satisfies the property (( ) ( )) ( ( )) ( ( )).
M b where b ∈ B.