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A note on additively completely regular seminearrings

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Abstract

In the endeavour of obtaining semigroup theoretic analogues i.e., the analogues of structure theorems of completely regular semigroups in the setting of additively regular seminearrings we could obtain some results in Mukherjee et al. (Commun Algebra 45(12):5111–5122, 2017). But we could not obtain the analogue of (i) ‘A semigroup is Clifford if and only if it is strong semilattice of groups’ and (ii) ‘ A semigroup is completely regular if and only if it is a union of groups’. In Mukherjee et al. (Commun Algebra, https://doi.org/10.1080/00927872.2018.1524011) we could obtain the analogue of (i) for some restricted type of left (right) Clifford seminearrings. The main purpose of this paper is to complete the remaining task i.e., to obtain the analogue of (ii) for a class of additively completely regular seminearrings. In order to accomplish this we have characterized those seminearrings which are union of near-rings (zero-symmetric near-rings) in the class of additively completely regular seminearrings.

Introduction

Following Weinert [8] we call \((S,+,.)\) a seminearring if (1) \((S,+)\) is a semigroup (not necessarily commutative), (2) (S, .) is a semigroup (not necessarily commutative), (3) \((a+b).c=a.c+b.c\) for all a, b, \(c\in S\) (“right distributive law”). A (right) seminearring \((S,+,.)\) is said to be with zero if 0 is the additive identity of S and 0 satisfies the property \(0.a = 0\) for all \(a\in S\). A seminearring S is said to be zero-symmeric if S is a seminearring with 0 in which \(s.0= 0\) for each \(s \in S\). The seminearring considered in [7] was zero-symmetric. But throughout this paper, unless mentioned otherwise, the seminearring need not have to be with zero.

We have initiated our study of additively regular seminearrings in [5] and continued in [2, 3]. In [2], while studying additively completely regular seminearrings we could obtain a characterization of a left (right) completely regular seminearring as a bi-semilatticeFootnote 1 of left (right) completely simple seminearrings (cf. Theorems 2.23 and 2.24 of [2]) which is the semigroup theoretic analogue of “A semigroup is completely regular if and only if it is a semilattice of completely simple semigroups”. We could also obtain a characterization of left (right) Clifford seminearring as a bi-semilattice of near-rings (zero-symmetric near-rings) (cf. Corollaries 3.10 and 3.12 of [2]) which is the semigroup theoretic analogue of “A semigroup is Clifford if and only if it is a semilattice of groups”. We have also obtained in [3] the characterization of a restricted type of left (right) Clifford seminearrings either as strong bi-semilattice of near-rings (zero-symmetric near-rings) or as strong distributive lattice of near-rings (zero-symmetric near-rings) which is the semigroup theoretic analogue of “A semigroup is Clifford if and only if it is strong semilattice of groups”. Thus in [3] we could solve partly the question raised in the Concluding remark of [2]. But we could not obtain the characterization of unions of near-rings (zero-symmetric near-rings) in the class of additively completely regular seminearrings which is the semigroup theoretic analogue of “A semigroup is completely regular if and only if it is a union of groups” (cf. Concluding remark of [2]). The main purpose of the present paper is to make an attempt to complete this unfinished work, i.e., to characterize those seminearrings which can be decomposed as union of near-rings (zero-symmetric) in the class of additively completely regular seminearrings. Since in semigroup theory, the following three classes—the class of completely regular semigroups, the class of semigroups which are union of groups and the class of semigroups which are semilattice of completely simple semigroups coincide, to accomplish the above mentioned unfinished work the possible first move is to investigate whether the class of seminearrings which are bi-semilattice of left (right) completely simple seminearrings, i.e., the class of left (right) completely regular seminearrings (cf. Theorem 2.23 of [2]) serves our purpose or not. In this direction we obtain in Example 2.8 a seminearring \((S,+,.)\) which is union of near-rings but neither a left nor a right completely regular seminearring. In particular, S satisfies the hypothesis of Theorem 2.4 but neither of the properties (i) and (ii) of the respective theorem holds here. This motivates us to remove the following restriction: “ae\(\mathcal {J}^{+}\)ea for all \(a\in S\), for all \(e\in E^{+}(S)\)” (cf. Theorem 2.4) from the notion of left (right) completely regular seminearrings and to introduce a new notion called generalized left (right) completely regular seminearrings (cf. Definitions 2.9 and 2.11) which is finally proved to be our desired class of seminearrings.

On generalized completely regular seminearrings

Before going to introduce the notion of generalized left (right) completely regular seminearrings we recall some preliminaries of left (right) complete regularity of seminearrings. We assume that a good extent of the semigroup theoretic background of seminearrings is well-known from [9].

Definition 2.1

[1, 2] Suppose \((S,+,.)\) is a seminearring. Then S is said to be an additively regular seminearring (additively inverse seminearring) if \((S,+)\) is a regular semigroup (resp. an inverse semigroup). Similar are the definitions of additively completely regularFootnote 2 and additively completely simple seminearrings (see footnote 2).

Notations 2.2

Throughout this paper, unless mentioned otherwise, (i) for a seminearring S, \(E^+(S)\) denotes the set of all additive idempotents; (ii) in an additively completely regular seminearring S, for \(a\in S\), an element \(x\in S\) satisfying \(a+x+a=a\) and \(a+x=x+a\) (p-58 [4], Definition 2.2) is denoted by \(x_{a}\); (iii) \(\mathcal {L}^+\), \(\mathcal {R}^+\), \(\mathcal {H}^+\) and \(\mathcal {J}^+\) denote the Green’s relations \(\mathcal {L}\), \(\mathcal {R}\), \(\mathcal {H}\) and \(\mathcal {J}\) on the semigroup \((S, +)\), the additive reduct of the seminearring S; (iv) in a seminearring S, \(\mathcal {L}^+_a\), \(\mathcal {R}^+_a\), \(\mathcal {H}^+_a\) and \(\mathcal {J}^+_a\) denote the \(\mathcal {L}^+\), \(\mathcal {R}^+\), \(\mathcal {H}^+\) and \(\mathcal {J}^+\) classes of \(a\in S\); (v) in an additively completely regular seminearring S, for each \(a\in S\), \((\mathcal {H}_{a}^+,+)\) is a group. The identity element of this group is denoted by \(0_{\mathcal {H}_{a}^+}\).

If \(\mathcal {L}^+\), \(\mathcal {R}^+\), \(\mathcal {H}^+\) and \(\mathcal {J}^+\) are congruences on the additive reduct of a semiring S, then they become congruences on the semiring S [6]. But the situation is not so nice in the case of a seminearring which is evident from the following example.

Example 2.3

[2] Let \((S,+)\) be a Clifford semigroup having at least three elements in which \(\mathcal {J}\) is neither trivial relation nor universal relation. Then M(S) is an additively Clifford seminearring and there exist three distinct elements a, b, \(c\in S\) such that a is \(\mathcal {J}\) related to b but not to c. Then there exist x, y, z, \(t\in S\) such that \(f_a = f_x\)\(+\)\(f_b\)\(+\)\(f_y\) and \(f_b = f_z + f_a + f_t\) where for \(t\in S\), \(f_t\in M(S)\) is defined by \(f_t(s)=t\) for all \(s\in S\). Thus \(f_a \mathcal {J}^+\)\(f_b\) in M(S). Let \(g\in M(S)\) such that \(g(a)=c\) and \(g(b)= b\). Then \(g\circ f_a\) is not related to \(g\circ f_b\) under \({\mathcal {J}}^+\) in M(S) though \(f_a\)\(\mathcal {J}^+\)\(f_b\). Therefore \(\mathcal {J}^+\) is not a congruence on the seminearring M(S) though \(\mathcal {J}^+\) is a congruence on \((M(S),+)\).

Theorem 2.4

[2] (With the same notation as in Notations 2.2 (ii)) Let S be an additively completely regular seminearring in which for every a there exists an \({x_{a}}\) satisfying \((a+x_{a})a=a+x_{a}\). Then the following are equivalent.   (i) For each \(a\in S\) and \(e\in E^{+}(S)\), ae\(\mathcal {J}^{+}\)ea. (ii) \(\mathcal {J}^{+}\) is a bi-semilattice congruence on S.

Theorem 2.5

[2] (With the same notation as in Theorem 2.4) Let S be an additively completely regular seminearring in which for every a there exists an \({x_{a}}\) satisfying \(a(a+x_{a})=a+x_{a}\). Then the following are equivalent.   (i) For each \(a\in S\) and \(e\in E^{+}(S)\), ae\(\mathcal {J}^{+}\)ea. (ii) \(\mathcal {J}^{+}\) is a bi-semilattice congruence on S.

Definition 2.6

[2] An additively completely regular seminearring \((S,+,.)\) satisfying the hypothesis of Theorem 2.4 (Theorem 2.5) is said to be a left (right) completely regular seminearring if it has any one of the properties (i) and (ii) of the respective theorems.

The following is a structure theorem for the above class of seminearrings which is analogue of “A semigroup is completely regular if and only if it is a semilattice of completely simple semigroups”

Theorem 2.7

[2] A seminearring is left (right) completely regular if and only if it is a bi-semilattice of left (right) completely simple seminearrings.

Now in order to complete the task of obtaining the analogue of “A semigroup is completely regular if and only if it is a union of groups” in the seminearring setting it is natural to determine the type of seminearings, in the class of additively regular seminearrings, which are union of near-rings. The following example motivates us to define (cf. Definition 2.9) a class of seminearrings which happens to be our desired class (cf. Theorem 2.19).

Example 2.8

Let us define ‘\(+\)’ on \(T=\{u,a,b,c\}\) as follows.

\(+\)uabc
uuabc
aaaaa
bbbbb
ccbau

\((T,+)\) is nothing but the full transformation semigroup on a set of two elements written in the additive notation. \((T,+)\) is a completely regular semigroup. Let S be the semigroup direct product of T with itself. Then clearly \((S,+)\) is again a completely regular semigroup. Let us define ‘\(*\)’ on S by \(x*y=x\) for all x, \(y\in S\). From the definition of ‘\(*\)’ and the fact that \((S,+)\) is completely regular it follows that S is union of near-rings. Clearly in S, \(\{(a,u), (b,u), (b,c), (a,c)\}\) and \(\{(a,b), (a,a), (b,b), (b,a)\}\) are two different \(\mathcal {J}^+\)-classes. Now as \((a,b)*(b,c)=(a,b)\) and \((b,c)*(a,b)=(b,c)\) belong to two different \(\mathcal {J}^+\)-classes, \((S,+,*)\) is not a left completely regular seminearring. The seminearring \((S,+,*)\) is additively regular and union of near-rings and satisfies the hypothesis of Theorem 2.4 but neither of the conditions (i), (ii) therein. Motivated by this we formulate the following definition.

Definition 2.9

(With the same notation as that of Notations 2.2(ii)) A seminearring \((S, +, .)\) is called generalized left completely regular (GLCR) if for each \(a \in S\) there exists an \(x_{a}\in S\) satisfying \((a + x_{a})a=a+ x_{a}\).

Remark 2.10

The defining condition in Definition 2.9 are nothing but the hypothesis of Theorem 2.4. Since Theorem 2.5 is the right analogue of Theorem 2.4, we formulate below the right analogue of Definition 2.9.

Definition 2.11

(With the same notation as that of Notations 2.2(ii)) A seminearring \((S, +, .)\) is called generalized right completely regular (GRCR) if for each \(a \in S\) there exists an \(x_{a}\in S\) satisfying \(a(a + x_{a})=a+ x_{a}\).

Thus in our project of studying additively regular seminearrings in order to obtain analogues of various structure Theorems of regular semigroups we have introduced so far four types of additively completely regular seminearring viz., generalized left completely regular (GLCR), generalized right completely regular (GRCR), left completely regular (LCR) [2] and right completely regular seminearrings (RCR) [2]. Regarding the relationships among these four notions of seminearrings it is relevent to ask some questions which we list below in a tabular form. Each row of the table contains one question. The way to read a question is explained in the footnote 3.Footnote 3 In the last column of the table we answer the questions and refer to supporting Examples/Remarks/Results.

Question LCRRCRGLCR GRCRAnswer
(i)\(\checkmark \)\(\checkmark \)\(\checkmark \)\(\checkmark \)Exists (cf. Remark 2.12 (ii))
(ii)\(\checkmark \)\(\checkmark \)\(\checkmark \)\(\times \)Does not exist (cf. Remark 2.12 (i))
(iii)\(\checkmark \)\(\checkmark \)\(\times \)\(\checkmark \)Does not exist (cf. Remark 2.12 (i))
(iv)\(\checkmark \)\(\times \)\(\checkmark \)\(\checkmark \)Does not exist (cf. Remark 2.12 (iii))
(v)\(\times \)\(\checkmark \)\(\checkmark \)\(\checkmark \)Does not exist (cf. Remark 2.12 (iii))
(vi)\(\checkmark \)\(\checkmark \)\(\times \)\(\times \)Does not exist (cf. Remark 2.12 (i))
(vii)\(\checkmark \)\(\times \)\(\checkmark \)\(\times \)Exists (cf. Example 2.14)
(viii)\(\checkmark \)\(\times \)\(\times \)\(\checkmark \)Does not exist (cf. Remark 2.12 (i))
(ix)\(\times \)\(\checkmark \)\(\checkmark \)\(\times \)Does not exist (cf. Remark 2.12 (i))
(x)\(\times \)\(\checkmark \)\(\times \)\(\checkmark \)Exists (cf. Example 2.16)
(xi)\(\times \)\(\times \)\(\checkmark \)\(\checkmark \)Exists (cf. Example 2.17)
(xii)\(\times \)\(\times \)\(\times \)\(\checkmark \)Exists (cf. Example 2.15)
(xiii)\(\times \)\(\times \)\(\checkmark \)\(\times \)Exists (cf. Example 2.8)
(xiv)\(\times \)\(\checkmark \)\(\times \)\(\times \)Does not exist (cf. Remark 2.12 (i))
(xv)\(\checkmark \)\(\times \)\(\times \)\(\times \)Does not exist (cf. Remark 2.12 (i))
(xvi)\(\times \)\(\times \)\(\times \)\(\times \)Exists (cf. Proposition 2.18)

The following remark is useful for finding answers to the Questions (i)–(vi), (viii), (ix), (xiv) and (xv).

Remark 2.12

  1. (i)

    If a seminearring S is LCR, then it is GLCR also and if a seminearring S is RCR, then it is GRCR also.

  2. (ii)

    Any zero-symmetric near-ring, being an LCR as well as an RCR seminearring, is a GLCR as well as a GRCR seminearring.

  3. (iii)

    If S is an LCR (RCR) as well as GRCR (respectively, GLCR), then it is RCR (respectively, LCR), too.

Example 2.13

The seminearring of Example 2.8 is a GLCR but neither LCR nor GRCR nor RCR.

The following example provides the answer to the Question (vii).

Example 2.14

Seminearrings cited in Example 2.16 of [2] are LCR and hence GLCR, but neither GRCR nor RCR.

The following example provides the answer to the Question (xii).

Example 2.15

Let us consider the semigroup \((S,+)\) considered in Example 2.8 which is completely regular. Let us consider the subseminearring \(IF(S):=\{f\in M(S)| f(e)=e \text { for all } e\in E(S)\}\) of \((M(S),+,\circ )\). Let \(f,g,h\in IF(S)\) be such that \(f(b,c)=(b,u)=f(c,b)\), \(g(a,c)=(b,c)\), \(h(a,c)=(c,b)\). Then \((f\circ (g+h))(a,c)\)\(\ne \)\((f\circ g+f\circ h)(a,c)\) whence \((IF(S),+,\circ )\) is not a left distributive seminearring.

Let \(f\in IF(S)\). Since \((S,+)\) is completely regular, for each \(f(a)\in S\setminus E(S)\) there exists \(x_{f(a)}\in S\) satisfying \(f(a)+x_{f(a)}+f(a)\)\(=\)f(a) and \(f(a)+x_{f(a)}= x_{f(a)}+f(a)\). Using the axiom of choice we define \(x_{f}:S\rightarrow S\) by

$$\begin{aligned} x_{f}(a)=\left\{ \begin{array}{ll} x_{f(a)},&{}\quad {\text {if}}\,\, f(a)\not \in E(S) \\ f(a),&{}\quad {\text {if}}\,\, f(a)\in E(S). \\ \end{array} \right. \end{aligned}$$

Clearly, \(x_{f}\in IF(S)\), \(f+x_{f}+f=f\), \(f+x_{f}=x_{f}+f\) and \(f\circ (f+x_{f})=f+x_{f}\). So \((IF(S),+,\circ )\) is a GRCR seminearring.

The \(\mathcal {H}\)-class of (uu) is \(\{(u,u),(c,u),(u,c),(c,c)\}\) and the \(\mathcal {H}\)-class of (ua) is \(\{(u,a),(c,a)\}\). Let us choose f from IF(S) so that \(f(c,u)=(u,c)\) and \(f(u,c)=(c,a)\). If IF(S) is a GLCR seminearing then f(a) \(\mathcal {H}\)\(f^{2}(a)\) for all \(a\in S\) (cf. Lemma 2.21) which contradicts the fact that f(cu), \((f\circ f)(c,u)\) lie in different \(\mathcal {H}\) classes of S. Hence \((IF(S),+,\circ )\) is not a GLCR.

In S, \(\{(a,u), (b,u), (b,c), (a,c)\}\) and \(\{(a,b), (a,a), (b,b), (b,a)\}\) are two different \(\mathcal {J}\) classes. Let us construct two functions \(g,h:S\rightarrow S\) in the following manner.

$$\begin{aligned}&g(x,y)=\left\{ \begin{array}{ll} (a,u),&{}\quad {\text {if }}\,\,(x,y)=(c,c)\\ (x,y),&{}\quad {\text {where}}\,\, x,y\in E(S) \\ (u,y),&{}\quad {\text {where}}\,\, x=c\, \mathrm{and}\, y\in E(S)\\ (x,u),&{}\quad {\text {where}}\,\, y=c\, \mathrm{and} \, x\in E(S)\\ \end{array} \right. \quad \\&h(x,y)=\left\{ \begin{array}{ll} (a,b),&{}\quad {\text {if}}\,\, (x,y)=(c,c) \\ (x,y),&{}\quad {\text {if}}\,\, (x,y)\ne (c,c). \\ \end{array} \right. \end{aligned}$$

Then clearly \(h\in IF(S)\) and \(g\in E^{+}(IF(S))\). Now \((g\circ h)(c,c)= (a,b)\) and \((h\circ g)(c,c)=(a,u)\). Consequently, \(g\circ h\) and \(h\circ g\) are not \(\mathcal {J}^{+}\) related in IF(S). Hence IF(S) is a GRCR seminearring which is neither a RCR nor GLCR seminearring.

The following example provides the answer to the Question (x).

Example 2.16

The seminearring IF(S) considered in Example 2.19 of [2] is an RCR seminearring (as well as GRCR) which is not an LCR seminearring. In view of Remark 2.12(iii), IF(S) is not also GLCR.

The following example provides the answer to the Question (xi).

Example 2.17

Let us consider a semilattice \((L,+)\) having at least two elements. Let us define ‘\(*\)’ on L by \(x*y=x\) for all x, \(y\in L\). It is a matter of routine verification that \((L,+,*)\) is a GLCR as well as a GRCR seminearring which is also a semiring. Now let \(a,b\in L\) such that \(a\ne b\). Then as \(a*b=a\) and \(b*a=b\), being two different additive idempotents, belong to two different \(\mathcal {J}^+\)-classes of \((L,+,*)\). Hence \((L,+,*)\) is neither an LCR nor an RCR seminearring.

The following proposition, the proof of which being straightforward is omitted, together with Remark 2.12 provides the answer to the Question (xvi).

Proposition 2.18

Let \(S_{L}\) be an LCR seminearring which is not RCR and \(S_{R}\) be an RCR seminearring which is not LCR. Then \(S_{L}\times S_{R}\), the seminearring direct product of \(S_{L}\) and \(S_{R}\) is neither GLCR nor GRCR.

Theorem 2.19

Let S be a seminearring. Then the following statements are equivalent:

  1. (1)

    S is generalized left completely regular (GLCR);

  2. (2)

    Every \(\mathcal {H}^+\)-class is a near-ring;

  3. (3)

    S is a union (disjoint) of near-rings.

In order to prove this theorem we need the following lemmas.

Lemma 2.20

Let S be a generalized left completely regular (GLCR) seminearring. Then \(\mathcal {H}^+\) is a right congruence (i.e., an equivalence relation which is compatible w.r.t. addition and right compatible w.r.t. multiplication) on S.

Proof

The proof follows immediately using right distributivity of the seminearring under consideration. \(\square \)

Lemma 2.21

Let S be a generalized left completely regular (GLCR) seminearring. Then \(a^2\mathcal {H}^+a\) for all \(a \in S\).

Proof

Let \(a\in S\). Then \(a =a +x_{a}+a = a + (x_{a}+a)a = (a + x_{a}a)+ a^2\) and \(a^2 = (a+x_{a}+a)a = a^2 + (a+x_{a})a = a^2 + x_{a} + a\). This implies \(a^2\)\(\mathcal {L}^+\)a. Again, \(a =(a+ x_{a})a +a = a^2 +(x_{a}a+a)\). Also \(a^2 = (a+x_{a}+a)a = (a+ x_{a})a + a^2 = a+ (x_{a}+a^2)\). Thus \(a^2 \mathcal {R}^{+} a\). Consequently, \(a^2 \mathcal {H}^{+} a\). \(\square \)

Lemma 2.22

Let S be a generalized left completely regular (GLCR) seminearring and \(a \in S \). Then for each \(y \in \mathcal {H}_{a}^+\) (cf. Notations 2.2(iv)) there exists a unique element \(y^* \in \mathcal {H}_{a}^+\) such that \(y + y^* + y\)\(=\)y, \(y + y^*\)\(=\)\(y^* + y\) and \((y + y^*)y\)\(=\)\(y + y^*\).

Proof

Let \(y \in \mathcal {H}_{a}^+\). Since \(\mathcal {H}_{a}^+\) is a group, there exists a unique \(y^*\in \mathcal {H}_{a}^+\) such that \(y + y^* = y^* + y = 0_{\mathcal {H}_{a}^+}\) (cf. Notations 2.2(v)) \(= e\), say and \(y + y^* + y = y\). Then \((y + y^*)y = ey \in E^+(S)\). Now \(y \mathcal {H}^+ e\). As \(\mathcal {H}^+\) is a right congruence (cf. Lemma 2.20), \(y^2 \mathcal {H}^+ ey\). Since \(y^2 \mathcal {H}^+ y\) (cf. Lemma 2.21), \(y \mathcal {H}^+ ey\). Hence \(ey \in \mathcal {H}_{a}^+\bigcap E^+(S)\). So, \(ey = e\). Thus \((y+y^*)y = e = y+y^*\). \(\square \)

Proof of the Main Theorem

\((1)\Rightarrow (2)\) Let \(a\in S\) and \(b, c \in \mathcal {H}_{a}^+\). Then in view of Lemma 2.22, there exist unique \(b^*\), \(c^*\in \mathcal {H}_{a}^+\) such that \(b+b^*+b=b\), \(c+c^*+c = c\), \(b+b^*=b^*+b=c+c^*=c^*+c\), \((b+b^*)b=b+b^*\), and \((c+c^*)c=c+c^*\). Then \(bc= (b+b^*)c + bc=(c+c^*)c +bc= c+(c^*+bc)\) and \(c = (c+c^*)c+ c = (b+b^*)c+c = bc+(b^*c+c)\). Hence \(bc \mathcal {R}^+ c\). Again, \(bc= bc + (b+b^*)c= bc + (c+c^*)c= (bc +c^*) + c\). Also, \(c= c+c+c^*= c+ (c+c^*)c= c + (b+b^*)c= (c+ b^*c)+bc\). Hence \(bc \mathcal {H}^+ c\). Then \(bc \mathcal {H}^+a\). Consequently, (\(\mathcal {H}_{a}^+\), +, .) is a near-ring.

\((2)\Rightarrow (3)\) Obvious.

\((3)\Rightarrow (1)\) Let S be a union of near-rings \(\{N_{\alpha }:\alpha \in \Lambda \}\) and \(a\in S\). Then \(a\in N_{\alpha }\) for some \(\alpha \in \Lambda \). Let \(a^*\) be the unique additive inverse of a in the near-ring \((N_{\alpha },+,.)\). Then clearly \(a+a^*+a\)\(=\)a, \(a+a^*\)\(=\)\(a^*+a\) and \((a+a^*)a\)\(=\)\(a+a^*\). So S is generalized left completely regular. \(\square \)

Remark 2.23

The right analogue of the above theorem does not hold, i.e., if S is a generalized right completely regular (GRCR) seminearring then it need not be decomposed as union (disjoint) of near-rings which is evident from the following example.

Example 2.24

Let us consider the seminearring \((IF(S), +, \circ )\) considered in Example 2.15 which is generalized right completely regular. Let us choose f from IF(S) so that \(f(c,u)=(u,c)\) and \(f(u,c)=(c,a)\). Then f(cu) and \((f\circ f)(c,u)\) lie in different \(\mathcal {H}^+\) classes of S. So \(\mathcal {H}_{f}^+\) is not a near-ring as it does not contain \(f^2\). Hence IF(S) can not be decomposed as union of near-ring in view of Theorem 2.19.

In the following result we characterize those seminearrings which are unions of zero-symmetric near-rings in the class of additively completely regular seminearrings.

Theorem 2.25

Let S be a seminearring. Then the following statements are equivalent:

  1. (1)

    S is generalized left (GLCR) as well as generalized right completely regular (GRCR);

  2. (2)

    Every \(\mathcal {H}^+\)-class is a zero-symmetric near-ring;

  3. (3)

    S is a union (disjoint) of zero-symmetric near-rings.

Proof

\((1)\Rightarrow (2)\) Suppose (1) holds. Let \(a\in S\). Then by Theorem 2.19, \(\mathcal {H}_{a}^+\) is a near-ring. So there exists \(a^*\in \mathcal {H}_{a}^+\) such that \(a+a^*\)\(=\)\(a^*+a\)\(=\)\(0_{\mathcal {H}_{a}^+}\). Again by Definition 2.11, there exists \(x_{a}\in S\) such that \(a+ x_{a}+a=a\), \(a+x_{a}=x_{a}+a\) and \(a(a+x_{a})=a+x_{a}\). Now \(a+x_{a}\)\(=\)\(x_{a}+a+0_{\mathcal {H}_{a}^+}\)\(=\)\(a+x_{a}+a+a^*\)\(=\)\(a+a^*\)\(=\)\(0_{\mathcal {H}_{a}^+}\). Hence \(a\cdot 0_{\mathcal {H}_{a}^+}\)\(=\)\(0_{\mathcal {H}_{a}^+}\). Consequently, \(\mathcal {H}_{a}^+\) is a zero-symmetric near-ring.

\((2)\Rightarrow (3)\) Obvious.

\((3)\Rightarrow (1)\) The proof that S is GLCR is the same as that of \((3)\Rightarrow (1)\) of Theorem 2.19. This together with the zero symmetricity implies that S is GRCR. \(\square \)

Concluding remark

Some parts i. e., (A), (B), (C) of Theorem 1.4 of [6] follows from our Theorem 2.25 if we replace the seminearring S by a semiring.Footnote 4

Notes

  1. 1.

    Bi-semilattice is a suitable substitute of semilattice in the setting of seminearring.

  2. 2.

    For semigroup theoretic counterparts of these notations we refer to [4].

  3. 3.

    \(\checkmark \) denotes the existence and \(\times \) denotes nonexistence. Thus the question asked in the \(2^{nd}\) row is—“Does there exist a seminearring which is RCR as well as GLCR as well as LCR but not GRCR?”

  4. 4.

    It may be noted that any semiring is also a seminearring.

References

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Acknowledgements

The authors are grateful to Prof. M. K. Sen of University of Calcutta for suggesting the problem. They are also grateful to Prof. S. K. Sardar of Jadavpur University and to Prof. Sen for their active guidance throughout the preparation of the paper. The authors are also grateful to the learned referee for meticulous review and subsequent suggestions for overall improvement of the paper.

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Correspondence to Rajlaxmi Mukherjee.

Additional information

Communicated by Lev N. Shevrin.

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Mukherjee, R., Manna, T. & Pal, P. A note on additively completely regular seminearrings. Semigroup Forum 100, 339–347 (2020). https://doi.org/10.1007/s00233-018-9983-9

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Keywords

  • Left (right) completely regular seminearring
  • Generalized left (right) completely regular seminearring
  • Union of near-rings (zero-symmetric near-rings)