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Kyungpook Mathematical Journal 2022; 62(1): 29-41

Published online March 31, 2022 https://doi.org/10.5666/KMJ.2022.62.1.29

Copyright © Kyungpook Mathematical Journal.

Left Regular and Left Weakly Regular n-ary Semigroups

Patchara Pornsurat and Bundit Pibaljommee*

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
e-mail : p.patchara@kkumail.com and banpib@kku.ac.th

Received: July 13, 2021; Revised: November 2, 2021; Accepted: November 15, 2021

We study the concept of a quasi-ideal and a generalized bi-ideal of an n-ary semigroup; give a construction of the quasi-ideal of an n-ary semigroup generated by its nonempty subset; and introduce the notions of regularities, namely, a left regularity and a left weakly regularity. Moreover, the notions of a right regularity, a right weak regularity and a complete regularity are given. Finally, characterizations of these regularities are presented.

Keywords: n-ary semigroup, left regular, right regular, quasi-ideal

The concept of a semigroup has been extensively studied for a long time, as has that of an ordered semigroups, which is a semigroup together with a partially ordered relation. Classical results on ordered semigroups can be found in [1, 2]. Many kinds of regularity of ordered semigroups were studied by Kehayopulu and Tsingelis (see, [12, 13, 14, 15, 16]). In 1991, Kehayopulu [13] introduced the concept of right regular ordered semigroups and considered semigroups in which the right ideals are two-sided. Kehayopulu [14] gave the concept of complete regularity and gave characterizations of completely regular poe-semigroups which generalize the characterizations of completely regular semigroups (without order) which is given by Steinfeld [24]. Additionally, the characterizations of right weakly regular semigroups can be found in [10, 17].

The definition and theory of ternary semigroups were first introduced by Lehmer [18] in 1932. Santiago [22] investigated the notion of ideals of ternary semigroups and gave some properties of regular ternary semigroups. In 2012, Daddi and Pawar [3] investigated the notion of ordered quasi-ideals and ordered bi-ideals of ordered ternary semigroups and used them to characterize regular ordered ternary semigroups. N. Lekkoksung and P. Jampachon [19] gave the other types of regularities of ordered ternary semigroups, namely, left (right) weakly regular ordered ternary semigroups. Later, several types of regularities of ordered ternary semigroups were characterized in terms of ordered ideals by Pornsurat and Pibaljommee in [7].

The notion of n-ary systems was first introduced by Kasner [11] in 1904. In 1928, Dörnte [5] studied the notion of n-ary groups which is a generalization of groups. Sioson [23] gave the notion of regular n-ary semigroups and used their ideals to characterize regular n-ary semigroups. Later, Dudek and Groździńska [6]gave a new definition of regular n-ary semigroups and investigated some properties of regular n-ary semigroups. Additionally, Dudek proved several results and gave many examples of n-ary groups in [7, 8]. Moreover, Dudek [9] studied a new type of elements of n-ary semigroups where n ≥ 3, namely, potent elements and investigated properties of ideals in which all elements are potent. Recently, the construction of the j-ideal of an n-ary semihypergroup generated by its nonempty subset was presented by Daengsaen, Leeratanavalee and Davvaz [4]. Later, Pornsurat, Palakawong na Ayutthaya and Pibaljommee 6 gave a construction of the j-ideal of a ordered n-ary semigroup generated by its nonempty subset in a different form.

In this work, we first study the concept of a quasi-ideal and a generalized bi-ideal of an n-ary semigroup and give a construction of the quasi-ideal of an n-ary semigroup generated by its nonempty subset. Then we introduce the notions of regularities, namely, a left regularity, a right regularity, a left weak regularity, a right weak regularity and a complete regularity in an n-ary semigroup. Moreover, we give characterizations of these regularities using properties of their ideals.

Let be the set of all natural numbers and i,j,n. A nonempty set S together with an n-ary operation given by f:SnS, where n ≥ 2, is called an n-ary groupoid [9]. For 1i<jn, the sequence xi,xi+1,xi+2,,xj of elements of S is denoted by xij and if xi=xi+1==xj=x, we write xji+1 instead of xij. For j<i, we denote xij as an empty symbol and so is x0. Then,

f(x1,,xi, x,x,,xjterms,xi+j+1,,xn)=f(x1i,xj,xi+j+1n).

An n-ary groupoid satisfies (i,j)-associative law if

f(x1i1,f(xin+i1),xn+i2n1)=f(x1j1,f(xjn+j1),xn+j2n1)

for all x1,,x2n1S. The n-ary operation f is associative if the above identity holds for all 1ijn. An n-ary groupoid (S, f) is called an n-ary semigroup if the n-ary operation f satisfies the associative law.

By the associativity of an n-ary semigroup, we define the map fk where k ≥ 2 by

fk(x1k(n1)+1)= f(f(,f(fkterms(x1n),xn+12n1),),x(k1)(n1)+2k(n1)+1)

for any x1,,xk(n1)+1S.

For 1i<jn, the sequence Ai,Ai+1,,Aj of nonempty sets of S is denoted by the symbol Aij. For nonempty subsets A1n of S, we denote

f(A1n)={f(a1n)aiAiwhere1in}.

If A1=A2==An=A, then we write f(An) instead of f(A1n). If A1={a1}, then we write f(a1,A2n) instead of f({a1},A2n), and similarly in other cases. For j<i , we set the notations Aij and A0 to be the empty symbols as a similar way of the notations xij and x0, respectively.

Throughout this paper, we write S instead of an n-ary semigroup. A nonempty subset H of S is called an n-ary subsemigroup of S if f(a1n)H for all a1,a2,,anH.

Many notations above can be found in [6, 7, 8, 9].

For 1in, a nonempty subset I of S is called an i-ideal of S if f(Si1,I,Sni)I. A nonempty subset I of S is called an ideal of S if I is an i-ideal for all i=1,,n.

Let A be a nonempty subset of S. The intersection of all i-ideals of S containing A is called the i-ideal of S generated by A. Then, we denote the notation Ji(A) to be the i-ideal of S generated by A. In a particular case A={a}, we write Ji(a) instead of Ji({a}).

As a special case of n-ary semihypergroups [4] and ordered n-ary semigroups [21], we present the construction of the i-ideal of an n-ary semigroup generated by its nonempty subset as the following theorem.

Theorem 2.1. Let A be a nonempty subset of S. Then

Ji(A)= m=1 n1fm(Sm(i1),A,Sm(ni))A.

Corollary 2.2. Let A be a nonempty subset of S. Then the following statements hold:

  • (i) J1(A)=f(A,Sn1)A and Jn(A)=f(Sn1,A)A.

  • (ii) If 1<i<n and i=n+12, then

Ji(A)=f(Si1,A,Sni)f2(Sn1,A,Sn1)A.

Next, we define the notion of a quasi-ideal of an n-ary semigroup and give a construction of the quasi-ideal of an n-ary semigroup generated by its nonempty subset as follows.

Definition 2.3. A nonempty subset Q of S is called a quasi-ideal of S if

i=1nm=1n1fm(Sm(i1),Q,Sm(ni))Q.

Note that for any quasi-ideal Q of S, f(Qn) m=1 n1fm(Sm(i1),Q,Sm(ni)) for all i=1,,n. So, f(Qn) i=1 nm=1n1fm(Sm(i1),Q,Sm(ni))Q. So, every quasi-ideal of S is an n-ary subsemigroup of S.

Proposition 2.4. Let S be an n-ary semigroup. Then the following statements hold.

  • (i) Every i-ideal of S is a quasi-ideal of S for each 1in.

  • (ii) If I1, I2,…,In are 1-ideal, 2-ideal,…, n-ideal of S, respectively, then i=1nIi is a quasi-ideal of S.

Proof. (1): Let I be an i-ideal of S. By Theorem 2.1,

i=1nm=1n1fm(Sm(i1),I,Sm(ni)) m=1 n1fm(Sm(i1),I,Sm(ni))Ji(I)=I.

So, every i-ideal of S is a quasi-ideal of S.

(2): Let Ii be an i-ideal for all 1in, respectively. Let Q= i=1nIi. Since Ii, there is aiIi for all i=1,,n. So, f(a1n)Q. By Theorem 2.1,

i=1nm=1n1 fm(Sm(i1),Q,Sm(ni))i=1nm=1n1 fm(Sm(i1),Ii,Sm(ni))          i=1nJi(Ii)          =i=1nIi.

So, Q is a quasi-ideal of S.

Let S be a semigroup. Then we note that the union of two quasi-ideal need not to be a quai-ideal of S as the following example.

Example 2.5. Let S={a,b,c,d,e,f,g,h}. Define the binary operation ⋅ on S by the following table:

Then (S,) is a semigroup. It is easy to see that {a, c} and {a, b} are quasi-ideal of S. However, {a, b, c} is not a quasi-ideal of S because {a,b,c}SS{a,b,c}={a,b,g}{a,b,c}.

Proposition 2.6. A nonempty subset of S is a quasi-ideal of S if and only if it is an intersection of a 1-ideal I1, a 2-ideal I2,…, an n-ideal In of S.

Proof. Let Q be a quasi-ideal of S. Define Ii= m=1 n1fm(Sm(i1),Q,Sm(ni))Q. By Theorem 2.1, Ii is an i-ideal of S for all i=1,,n. It is clear that Q i=1nIi. We consider i=1nIi= i=1nm=1n1fm(Sm(i1),Q,Sm(ni))Q. So, Q= i=1nIi. The converse is followed by Proposition 2.4(2).

For a nonempty subset A of S, we denote the notion Q(A) to be the quasi-ideal of S generated by A. In a particular case A={a}, we write Q(a) instead of Q({a}).

Theorem 2.7. The intersection of an arbitrary nonempty family of quasi-ideal of S is either a quasi-ideal of S or an empty set.

Proof. Let {QjjJ} be a family of quasi-ideal of S. Suppose that Q= jJQj. Since Qj is a quasi-ideal of S for all j∈ J, i=1nm=1n1fm(Sm(i1),Q,Sm(ni)) i=1nm=1n1fm(Sm(i1),Qj,Sm(ni))Qj for all j∈ J. So, i=1nm=1n1fm(Sm(i1),Q,Sm(ni)) jJQj=Q. Hence, Q is a quasi-ideal of S.

Theorem 2.8. Let S be an n-ary semigroup. Then

Q(A)= i=1nm=1n1fm(Sm(i1),A,Sm(ni))A.

Proof. Let A be a nonempty subset of S. By Theorem 2.1 and Proposition 2.4(2), i=1nm=1n1fm(Sm(i1),A,Sm(ni))A is a quasi-ideal of S containing A. So, Q(A) i=1nm=1n1fm(Sm(i1),A,Sm(ni))A. Let Q be a quasi-ideal of S and AQ. Then

i=1nm=1n1 fm(Sm(i1),A,Sm(ni))Ai=1nm=1n1 fm(Sm(i1),Q,Sm(ni))Q          QQ          =Q.

Therefore, Q(A)= i=1nm=1n1fm(Sm(i1),A,Sm(ni))A.

Definition 2.9. A nonempty subset B of S is called a generalized bi-ideal of S if f2( B,S,B,,S,B2n1terms)B.

Remark 2.10. Every quasi-ideal of S is a generalized bi-ideal of S.

Proof. Let B be a quasi-ideal of S. We consider

f2( B,S,B,,S,B2n1terms)f2(S2(i1),B,S2(ni)) m=1 n1fm(Sm(i1),A,Sm(ni))

for all i=1,,n.

It follows that, f2( B,S,B,,S,B2n1terms) i=1nm=1n1fm(Sm(i1),B,Sm(ni))B.

Now, we recall the notion of a regular n-ary semigroup in sense of Sioson as follows.

An n-ary semigroup S is called regular, in sense of Sioson [23], if each a ∈ S, there exist xijS where i,j=1,2,n such that

f(f(a,x121n),f(x21,a,x232n),,f(xn1nn1,a))=a.

Many forms of regular n-ary semigroup which can also be found in [6] are given as follows.

An n-ary semigroup S is called regular if each a ∈ S, there exist x2,,xnS such that

f2(a,x2,a,x3,,a,xn,a)=a.

An n-ary semigroup S is called regular if each a∈ S, there exist x2,,x2n2S such that

f2(a,x22n2,a)=a.

An n-ary semigroup S is called regular if each a∈ S, there exist x2,,xn1S such that

f(a,x2n1,a)=a.

However, Dudek and Groździńska [6] proved that the regularities conditions (2.1), (2.2), (2.3) and (2.4) are equivalent.

In this section, we define the notion of left regular, right regular, left weakly regular, right weakly regular and completely regular n-ary semigroups and give some their characterizations.

Definition 3.1. An element a∈ S is called a left (right) regular element if there exist x1,,xn1S such that a=f2(x1n1,an) (a=f2(an,x1n1)). An n-ary semigroup S is left (right) regular, if each its element is left (right) regular.

We note that S is a left (right) regular n-ary semigroup where n ≥ 3 if and only if there exists x∈ S such that a=f(x,an1) (a=f(an1,x)).

Lemma 3.2. The following statements are equivalent.

  • (i) S is left (resp. right) regular.

  • (ii) Af2(Sn1,An) (resp. Af2(An,Sn1)) for any AS.

  • (iii) af2(Sn1,an) (resp. af2(an,Sn1)) for any a ∈ S.

Definition 3.3. Let P be a nonempty subset of S. Then P is called semiprime if for all nonempty subsets A of S, f(An)P implies AP.

Remark 3.4. A nonempty subset P of S is semiprime if and only if for all a ∈ S, f(an)P implies a ∈ P.

Theorem 3.5. The following statements are equivalent.

  • (i) S is left regular.

  • (ii) Every n-ideal of S is semiprime.

  • (iii) Jn(a) is a semiprime of S for any a∈ S.

  • (iv) Jn(f(an)) is a semiprime of S for any a ∈ S.

Proof. (i)(ii). Let I be an n-ideal of S and AS such that f(An)I. Since S is left regular and By Lemma 3.2, Af2(Sn1,An)f(Sn1,I)I.

(ii)(iii) and (iii)(iv) are obvious.

(iv)(i). Let aS. Since f(an)Jn(f(an)) and Jn(f(an)) is semiprime, we have that aJn(f(an)). By Corollary 2.2, aJn(f(an))=f(Sn1,f(an)){f(an)}. Then af(Sn1,(f(an)) or a{f(an)}. If a{f(an)}, then a=f(an)=f(an1,f(an))f(Sn1,f(an))=f2(Sn1,an). So, S is left regular.

Next theorem can be similarly proved as Theorem 3.5.

Theorem 3.6. The following statements are equivalent.

  • (i) S is right regular.

  • (ii) Every 1-ideal of S is semiprime.

  • (iii) J1(a) is a semiprime of S for any a∈ S.

  • (iv) J1(f(an)) is a semiprime of S for any a ∈ S.

Theorem 3.7. Let S be an n-ary semigroup. Then S is both left regular and right regular if and only if every quasi-ideal of S is semiprime.

Proof. Assume that S is both left regular and right regular. Let AS and Q be a quasi-ideal of S such that f(An)Q. By Lemma 3.2, Af2(An,Sn1)f(Q,Sn1) and Af2(Sn1,An)f(Sn1,Q). For i=2,,n1, we have that

Af2(An,Sn1)f2(f2(Sn1,An),An1,Sn1)=f4(Sn1,A2n1,Sn1)=f(f(Sn1,A),Ai2,f(An),f(Ani,Si),Sni1)f(Si1,Q,Sni).

Hence, Af(Si1,Q,Sni) for all i=1,,n. This implies that

A i=1nf(Si1,Q,Sni) i=1nm=1n1fm(Sm(i1),Q,Sm(ni))Q.

So, every quasi-ideal of S is semiprime. Conversely, assume that every quasi-ideal of S is semiprime and AS. By Theorem 2.8., f(An)Q(f(An))= i=1 nm=1n1fm(Sm(i1),f(An),Sm(ni))f(An). By assumption,

A i=1nm=1n1fm(Sm(i1),f(An),Sm(ni))f(An) m=1 n1fm(f(An),Sm(n1))f(An).

We consider

f(An)i=1 nm=1n1 fm(Sm(i1),f(An),Sm(ni))f(An)  m=1n1fm(f(An),Sm(n1))f(An)  m=1n1fm(f(An),Sm(n1))f(An1,m=1n1fm(f(An),Sm(n1))f(An))  =m=1n1fm(f(An),Sm(n1))m=1n1f m+1(An1,f(An),Sm(n1))f2(A2n1)  =m=1n1fm(f(An),Sm(n1))m=1n1fm(f(An),f(An1,S),Sm(n1)1)f2(A2n1)  m=1n1fm(f(An),Sm(n1))  =f2(An,Sn1).

We have that A m=1 n1fm(f(An),Sm(n1))f(An) m=1 n1fm(f(An),Sm(n1))f2(An,Sn1)=f2(An,Sn1)f2(An,Sn1)=f2(An,Sn1). Similarly, we can show that Af2(Sn1,An). Therefore, S is both left regular and right regular.

Definition 3.8. An n-ary semigroup S is called completely regular if it is regular, left regular and right regular.

Lemma 3.9. The following statements are equivalent.

  • (i) S is completely regular.

  • (ii) Af4(An, S,A,S,,A,S2n3terms,An) for any AS.

Proof. Assume that S is completely regular. Let AS. If n=2, then it is easy to see that Af2(A,S,A)f2(f2(A2,S),S,f2(S,A2))f4(A2,S,A2). If n ≥ 3, then

Af2(A,S,A,,S,A2n1terms)f2(f2(An,Sn1),S,A,S,,A,S2n3terms,f2(Sn1,An))=f2(f(An),f(Sn1,S),A,S,A,S,A2n5terms,f(Sn1,S),f(An))f2(f(An),S,A,S,A,S,A2n5terms,S,f(An))=f4(An,S,A,S,,A,S2n3terms,An).

Conversely, assume that Af4(An, S,A,S,,A,S2n3terms,An) for any AS. Let AS. It is not hard to see that Af4(An, S,A,S,,A,S2n3terms,An)f2(Sn1,An), Af4(An, S,A,S,,A,S2n3terms,An)f2(An,Sn1) and Af4(An, S,A,S,,A,S2n3terms,An)f(A,Sn2,A). So, S is completely regular.

For convenience, we write Af4(An, S,A,S,,A,S2n3terms,An)f(A,Sn2,A). instead of f4(An, S,A,S,,A,S2n3terms,An).

For n=2, we set f4(An,(S,An2),S,An)=f4(A2,S,A2).

Theorem 3.10. Let S be an n-ary semigroup. Then S is completely regular if and only if every generalized bi-ideal of S is semiprime.

Proof. Assume that S is completely regular. Let AS and B be a generalized bi-ideal of S such that f(An)B. By Lemma 3.9,

Af4(An,S,A,S,,A,S2n3terms,An)=f2(f(An),S,A,S,,A2n4terms,S,f(An))f2(f(An),S, f 2 (Sn1 , A n ),S,, f 2 (Sn1 , A n )2n4terms,S,f(An))=f2(f(An),f(S,Sn1 ),f( A n ),,f(S,Sn1 ),f( A n )2n4terms,S,f(An)f2(B,S,B,,S,B2n1terms)B.

So, B is semiprime. Conversely, assume that every generalized bi-ideal of S is semiprime. Let AS and B=f4(An,(S,An2),S,An). We will show that B is a generalized bi-ideal of S. We consider

f2(B,S,B,,S,B2n1terms)=f2(f4(An,(S,An2),S,An),S,B,,S,B,S2n3terms,f4(An,(S,An2),S,An))        =f4(An,(S,An2),f6(S,An,S,B,,S,B,S2n3terms,An,(S,An2),S),An)        f4(An,(S,An2),S,An)        =B.

So, B is a generalized bi-ideal of S. Next, we show that S is completely regular.

  • • If n=2, then it easy to see that f7(A8)f4(A2,S,A2). By assumption, f3(A4)f4(A2,S,A2),f(A2)f4(A2,S,A2) and so Af4(A2,S,A2). So, S is completely regular.

  • • If n=3, then it easy to see that f4(A9)f4(A3,S,A,S,A3). By assumption, f(A3)f4(A3,S,A,S,A3) and so Af4(A3,S,A,S,A3). So, S is completely regular.

  • • If n ≥ 4, then f( f(An),,f(An)nterms)=f5(An,An,An, f(An),,f(An)n4terms,An) =f5(An,A2n4,A4,f(An ),,f(An )n4terms,An)=f4(An,A2n4,f(A4,f(An ),,f(An )n4terms),An) f4(An,(S,An2),S,An). By assumption, f(An)f4(An,(S,An2),S,An) and so Af4(An,(S,An2),S,An). By Lemma 3.9., S is completely regular.

Definition 3.11. An n-ary semigroup S is called left (right) weakly regular if af( f(Sn1,a),,f(Sn1,a)nterms) (af( f(a,Sn1),,f(a,Sn1)nterms)) for all a ∈ S.

Remark 3.12. Let S be an n-ary semigroup.

  • (i) If S is left regular, then it is left weakly regular.

  • (ii) If S is right regular, then it is right weakly regular.

Theorem 3.13. The following statements are equivalent.

  • (i) S is left weakly regular.

  • (ii) L=f(Ln) for every n-ideal L of S.

Proof. Assume that S is left weakly regular. Let L be an n-ideal of S and a ∈ L. Then f(Sn1,a)L. Since S is left weakly regular, we have af( f(Sn1,a),,f(Sn1,a)nterms)f(Ln). So, Lf(Ln) It is clear that f(Ln)L.

Hence, L=f(Ln) for every n-ideal L of S. Conversely, assume that L=f(Ln) for every n-ideal L of S. Let a ∈ S. Then aJn(a)=f( J n (a),, J n (a)nterms), i.e., a=f(b1,,bn) where b1,,bnJn(a). By Corollary 2.2, Jn(a)=f(Sn1,a){a}. We have bn= a or bnf(Sn1,a) for i=1,,n. If bn=a, then a=f(b1,,bn)f(Sn1,a). If bnf(Sn1,a), then a=f(b1,,bn)f(b1,,bn1,f(Sn1,a))f(Sn1,a). This implies that af(Sn1,a). We consider Jn(a)=f(Sn1,a){a}f(Sn1,a). Thus, biJn(a)=f(Sn1,a) for all i=1,,n. We consider a=f(b1,,bn)f( f(Sn1,a),,f(Sn1,a)nterms). Therefore, S is left weakly regular.

The next theorem can be similarly proved as Theorem 3.13.

Theorem 3.14. The following statements are equivalent.

  • (i) S is right weakly regular.

  • (ii) R=f(Rn) for every 1-ideal R of S.

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