On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups

Abstract In this paper, we develop a technique which enables us to obtain several results from the theory of Γ-semigroups as logical implications of their semigroup theoretical analogues.


Introduction and preliminaries
The theory of Γ-semigroups has been around for more than three decades and counts hundreds of research papers and many PhD theses. Along with Γ-semigroups, other structures such as ordered Γ-semigroups and fuzzy Γ-semigroups have been studied in recent years. The majority of the results proved so far are Γ-analogues of the well-known results of ordinary semigroups which their authors pretend to be genuine generalizations of their semigroup counterparts. It should be noted that there is a striking similarity between the proofs of the original semigroup theorems and their Γ-semigroup analogues. It is this similarity that is causing a growing concern among Γ-skeptics that many of the results in Γ-semigroup theory are logically equivalent with their counterparts in ordinary semigroups. But so far there has been no evidence that this concern is mathematically based. The aim of this paper is to develop a technique whose purpose is to demonstrate the equivalence for a pair of analogue results from the two theories. This technique is a refinement of that developed in [1] and has the advantage that it works for regular Γ-semigroups endowed with a partial order. More specifically, given an ordered Γ-semigroup ( ≤ ) S, Γ, S , we construct an ordered semigroup ( ⋅ ≤ ) Ω , , γ Ωγ 0 0 and prove that S is regular if and only if Ω γ 0 is regular. This shows that regularity in the theory of Γ-semigroups can be interpreted as the usual regularity of semigroups. We go on further to prove that two characterizations of regularity, one for ordered Γ-semigroups and the other for ordered semigroups are logically equivalent. The characterization of the regularity of ordered Γ-semigroups is Theorem 8(iii) of [2] and also Theorem 3 of [3], which states that an ordered Γ-semigroup ( ≤ ) S, Γ, S is regular if and only if one-sided ideals of ( ≤ ) S, Γ, S are idempotent, and for every right ideal R and every left ideal L of ( ≤ ) S, Γ, S , ( ] R L Γ is a quasi ideal of ( ≤ ) S, Γ, S . On the other hand, the characterization of the regularity of ordered semigroups is Theorem 3.1(iii) of [4], which states that an ordered semigroup ( ⋅ ≤ ) S, , S is regular if and only if, one-sided ideals of ( ⋅ ≤ ) S, , S are idempotent, and for every right ideal R and every left ideal L of ( ≤ ) S, Γ, S , ( ] R L Γ is a quasi ideal of ( ⋅ ≤ ) S, , S . Proving that the above analogue theorems are equivalent gives points to the idea that producing Γ-analogues of known results from the semigroup theory brings nothing new to the theory as pretended, but simply replicates those results in a new setting.
In what follows, we give a few basic notions that will be used throughout the paper. Let S and Γ be two nonempty sets. Any map from × × S S Γ to S will be called a Γ-multiplication in S and is denoted by (⋅) Γ . The result of this multiplication for ∈ a b S , and ∈ γ Γ is denoted by aγb. In 1986, Sen and Saha [5,6] introduced the concept of a Γ-semigroup S as an ordered pair ( (⋅) ) S, Γ , where S and Γ are nonempty sets and (⋅) Γ is a Γ-multiplication on S, which satisfies the following property: Here we give some necessary definitions from ordered semigroup and ordered Γ-semigroup theory.

Construction of Ω γ 0
Given an ordered Γ-semigroup ( ≤ ) S, Γ, S , we define an ordered semigroup ( ⋅ ≤ ) Ω , , γ Ωγ 0 0 . To define Ω γ 0 we use the fact that we can always define a multiplication • on any nonempty set Γ in such a way that ( ) Γ;• becomes a group. This in fact is equivalent to the axiom of choice. Also, we use the concept of the free product of two semigroups. Material related to this concept can be found in [8, pp. 258-261]. Furthermore, let ( ⋅) F; be the free semigroup on S. Its elements are finite strings ( … ) x x , , n 1 , where each ∈ x S i and the product ⋅ is the concatenation of words. Now we define Ω γ 0 as the quotient semigroup of the free product * F Γ of ( ⋅) F; with ( ) Γ, • by the congruence generated from the set of relations a fixed element. The following is Lemma 2.1 of [9]. We have included here for convenience. Readers unfamiliar with rewriting systems can find anything necessary to understand the proof in the monograph [10]. Proof. First, we have to prove that the reduction system arising from the given presentation is Noetherian and confluent, and therefore any element of Ω γ 0 is given by a unique irreducible word from ∪ S Γ. Second, we have to prove that the irreducible words have one of these five forms. So if ω is a word of the form = ( ) ω u x γ y v , , , , for ∈ ∈ γ x y S Γ, , and u v , possibly empty words, then ω reduces to ′ = ( ) ω u xγy v , , . And if = ( ) ω u x y v , , , , then it reduces to ′ = ( ) ω u xγ y v , , 0 . In this way, we obtain a reduction system which is length reducing and therefore it is Noetherian. To prove that this system is confluent, from Newman's lemma, it is sufficient to prove that it is locally confluent. For this, we need to see only the overlapping pairs.
• , To complete the proof, we need to show that the irreducible word representing the element of Ω γ 0 has one of the five forms stated. If the word which has neither a prefix nor a suffix made entirely of letters from Γ, then it reduces to an element of S by performing the appropriate reductions. If the word has the form where ω is a word which has neither a prefix nor a suffix made entirely of letters from Γ, and α, ′ α have only letters from Γ, then it reduces to an element of one of the first three forms. □ Definition 2.1. We define an order relation ≤ Ωγ 0 in terms of ≤ S as follows: The restriction of the relation in Γ is taken to be the equality.
Using the fact that ≤ S is an order relation in the Γ-semigroup S, we can prove that ≤ Ωγ 0 is an order relation in the semigroup Ω γ 0 . It is obvious that ≤ Ωγ 0 is reflexive, and very easy to see that it is antisymmetric. We check for convenience the transitivity.
Next we prove that the compatibility of ≤ S in S implies that of ≤ Ωγ 0 in Ω γ 0 . We obtain the proof only for relations of type (4) of Definition 2.1 since the proofs for the other types are analogous. So let ′ ≤ ′ γxγ γyγ Ωγ 0 , and want to prove that the inequality is preserved after multiplying both sides of the above on the left (resp. on the right) by one of the following elements: Since the proofs for the compatibility on the right are symmetric to those on the left, we obtain them only for the left multiplication.
Therefore, ≤ Ωγ 0 is compatible with the multiplication of Ω γ 0 . Summarizing, we have the following.
Since in Section 3 we deal with ordered ideals in both structures, ( Ωγ 0 0 , we will not use the standard notation ( ] X to indicate the ordered ideal, but we introduce a new one as in the following definition.
, and for every ⊆ We remark by passing the following.
, then ≤ w c Ωγ 0 for ∈ c C and by Definition 2.1 we must have that ∈ w S, and that ≤ ω c S , proving that ∈ ω L S . □ The following lemma gives a relationship between the principal ordered ideal in S generated by some ∈ x S and the principal ordered ideal in Ω γ 0 generated by the same element x.

Lemma 2.3. Let ∈
x S by an arbitrary element. The following hold true.
(i) The principal left ordered ideal of Ω γ 0 generated by x is the set ( ) Proof. We obtain the proof for (i) since the proof for (ii) is dual to that of (i). So we have to prove that x Ω γ 0 , then B may have these forms: . One can see that in the same way as above, 3 Regularity in ordered Γ-semigroups as a consequence of regularity in ordered semigroups The following proposition shows that the regularity of an ordered Γ-semigroup can be completely characterized as the regularity of an ordered semigroup.

Proposition 3.1. S is a regular ordered Γ-semigroup if and only if Ω γ 0 is a regular ordered semigroup.
Proof. If S is a regular ordered Γ-semigroup, then for all ∈ ∃ ∈ a S x S , and ∈ γ γ , Γ 1 2 , such that ≤ a aγ xγ a S 1 2 . To prove Ω γ 0 is a regular ordered semigroup, we have to prove that every element of Ω γ 0 have an ordered inverse in Ω γ 0 . By Lemma 2.1, we have that the elements of Ω γ 0 can be represented by an irreducible word which has only five forms. We prove regularity for elements of each of these five forms. So let first ∈ α aα Ω γ . Hence, we showed that Ω γ 0 is a regular ordered semigroup. Conversely, if Ω γ 0 is regular ordered semigroup, then every ∈ a S has an inverse in Ω γ 0 . To show that S is a regular ordered Γ-semigroup, we show that very ∈ a S has an inverse in S. For this, we distinguish between the five following forms. First, if the inverse of a in Ω γ 0 has the form ∈ αxβ Ω γ 0 , for ∈ x S, then ≤ a aαxβa is a quasi ideal, we have

S S S S S S S S S S S
From this and the previous assumptions, we obtain Proof. ( ) ⇒ ( ) i ii is trivial since any regular ordered semigroup can be regarded as an regular ordered Γ-semigroup, where Γ is a singleton. Also, one-sided ideals and quasi ideals in ordered semigroups are the same as those in ordered Γ-semigroups when Γ is a singleton.

a S L S a L a S L a S L a S L S a L S a L S a L L L a S L S a S L S L L a S L S a L L a S L S a L a a
( ) ⇒ ( ) ii i . Assume first that the ordered Γ-semigroup S is regular. Then by Proposition 3.1, Ω γ 0 is a regular ordered semigroup. Theorem 3.2 of [4] implies that for every right ideal R and every left ideal L of is a quasi-ideal of Ω γ 0 , and also every right and left ideal of the semigroup Ω γ 0 is idempotent. Let A now be an ordered right ideal of S and consider the subset = ( ∪ ) is a right ideal of the ordered semigroup Ω γ 0 . To this end, we have to prove that it satisfies the two conditions of right ideals: . To show the first we have to show that for every ∈ ( ∪ ) b L A AΓ and this depends on the value of C. Since where the last inclusion comes from the fact that A is an ordered right ideal of ( ≤ ) S, Γ, S . It remains to prove that the same holds true in the second case when = ′ b a γ with ′ ∈ a S and ′ ≤ a a S , and ∈ γ Γ. Depending on the value of C we have to prove that ∈ ( ∪ ) aγC L A AΓ Ωγ 0 . Indeed, Ωγ 0 All the above verifications prove the first condition, while the second condition is obvious. So = R L Ωγ 0 ( ∪ ) A AΓ is a right ideal of the ordered semigroup Ω γ 0 and from [4] it follows that = ( ∪ ) R L A AΓ Ωγ 0 is an idempotent. Passing now from the ordered semigroup Ω γ 0 to the ordered Γ semigroup S, we show that if every right ideal of the ordered semigroup Ω γ 0 is idempotent, then so is every right ideal of the ordered Γ-semigroup S. Let A be a right ideal of the ordered Γ semigroup S, we have to prove A is an idempotent in S, that is, To prove the converse, we utilize the fact that This implies that every ∈ a A is lower with respect to ≤ Ωγ 0 than some element of A or some element of AΓ. The second case is impossible from the way we have defined ≤ Ωγ 0 , so it remains that there is some ∈ γ Γ, and ′ ″ ∈ a a A , such that ≤ ′ ″ a a γa is a quasi ideal of ( ≤ ) S, Γ, S , which means that: and From [4] we have that for the right ideal is a quasi ideal of Ω γ 0 . To prove the first condition (1), we see that where the last equality follows from Lemma 2.2. The second condition (2) is obviously true since ( ) Conversely, we assume that every right and left ideal of S is an idempotent, and for every right ideal A of S, and every left ideal B of S, the set ( ) is a quasi ideal of S, and want to prove that S is regular. The strategy is to prove that under the given conditions, Ω γ 0 is a regular ordered semigroup, and then from Proposition 3.1 we obtain straightaway that S is a regular Γ-semigroup. To prove the regularity of Ω γ 0 , it is enough to prove that all right ideals R and all left ideals L of Ω γ 0 are idempotent, and ( ⋅ ) L R L Ωγ 0 is a quasi ideal of Ω γ 0 . Let R be a right ideal of Ω γ 0 and want to prove that = RR R. The inclusion ⊆ RR R is trivial. To prove the reverse inclusion ⊆ R RR, we need to prove that every ∈ x R is of the form = which proves the claim.
(iv) The element of R is of the form αxβ. We first note that This shows that αxβ is expressed as a product of two elements of R.
(v) The final case is when the element of R is some ∈ γ Γ. Observe that Now letting 1 be the unit element of ( ) Γ, • we have that = ⋅ ∈ γ γ RR 1 , and once again, ∈ γ R is expressed as a product of two elements in R, namely, γ and 1.
Recollecting, we have proved that any right ideal R of Ω γ 0 is idempotent. Similarly, we can show that any left ideal L is idempotent in Ω γ 0 . Now if R is a right ideal and L a left ideal of the ordered semigroup Ω γ 0 , we have to prove that ( ) L RL Ωγ 0 is a quasi ideal of Ω γ 0 . This would follow immediately if we prove that ∩ ⊆ ( ) R L L RL . The remaining cases for an element from ∩ R L include elements of the form αx, xβ or simply x, where ∈ α β , Γ and ∈ x S. These cases are dealt similarly as above. □ The following is straightforward.
Corollary 3.1. Any of the characterizations of the regularity of an ordered Γ-semigroup given in Theorem 8 of [2] is logically equivalent to its corresponding characterization of the regularity of an ordered semigroup given in Theorem 3.1 of [4].