Some subvarieties of semirings variety COS${}_n^{•}$

1. Introduction

A semiring $(S,+,\cdot )$ is an algebra of type $(2,2)$ satisfying the following additional identities:

A variety of algebras is a class of algebras of the same type that is closed under the formation of subalgebras, homomorphic images and direct products (see [1]). A class, by Birkhoff’s theorem, of algebras of the same type is a variety if and only if it is an equational class. Obviously, all semirings form a variety.

Since semirings are two semigroups on the same nonempty set connected by distributive laws, the results and methods in algebraic theory of semigroups can be applied in the study of semirings (see [5,11]). Some scholars studied different completely regular semirings by Green’s relations of semigroups. Pastijn and Zhao [10] characterized idempotent semiring varieties and proved that Green’s ${{𝒟}}^{•}$-relation on an idempotent semiring is the least lattice congruence. Zhao et al. [18,17] studied idempotent semiring varieties using Green’s relations on additive reduct and multiplicative reduct. Damljanović et al. [4] obtained congruence openings of Green’s relations on the additive reduct of a semiring. Wang and Shao [13] showed that $S\in R_n\circ (N\cap \stackrel{+}{S\ell })$ is isomorphic to a subsemiring of $S/\sigma \times S/{{\mathscr{H}}}^{+}$. Cheng and Shao [2,3] defined and studied several varieties of multiplicative idempotent semirings by congruence openings of multiplicative Green’s relations on a semiring. Xian et al. [16] gave equivalent characterization of Green’s relations on the additive (respectively, multiplicative) reduct of the semiring in ${\mathrm{\text{COS}}}_3^{+}$ and obtained the sufficient and necessary conditions which make intersections and joins of these binary relations be congruences. Xian et al. [15] considered the closure congruences of Green’s relations on the additive (respectively, multiplicative) reduct of semirings, and several varieties of semirings determined by the closure congruences are characterized. Wang et al. [14] gave some equivalent descriptions of intersections between Green’s relations on the additive reduct and Green’s relations on multiplicative reduct of a semiring with additional identities $x^n\approx x$, $(2^n-1)x\approx x$, ${(x+y)}^{n-1}\approx x^{n-1}+y^{n-1}$ and ${(xy)}^{n-1}\approx x^{n-1}{y}^{n-1}$ and obtained the necessary conditions which make these binary relations be congruences.

With continuous research, some scholars have also paid attention to some semirings whose additive reducts are not completely regular. As a generalization of completely regular semiring, Maity and Ghosh [7, Theorem 3.10] defined quasi completely regular semiring, which is semiring $(S,+,\cdot )$ satisfying the following conditions: For every $a\in S$, there exists a positive integer n such that $na$ is completely regular, that is,

for a suitable element $x\in S$. Maity and Ghosh [12,7] showed that a semiring is quasi completely regular if and only if every ${{\mathscr{H}}}_{\ast }^{+}$-class is a quasi skew-ring. Moreover, they [8,9] investigated the congruence generated by a special equivalence relation on quasi completely regular semirings with and without the special conditions like being quasi-orthodox and characterized the least completely regular semiring congruence on quasi completely regular semirings.

In this paper, we denote by COS${}_n^{•}$ the semiring variety satisfying the following additional identities for $n\ge 2$:

For any $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a\in S$, it is easy to check $2a=2^{n}a$ and so $2a$ is a completely regular element in $(S,+)$. Hence, $(S,+)$ is completely $\pi$-regular (see [7]). Obviously, $(S,\cdot )$ is completely regular. Let $E_{+}(S)$ (respectively, $E_{•}(S)$) denote the set of all idempotents of $(S,+)$ (respectively, $(S,\cdot )$) and $m=2^n-2$. We can easily prove that $E_{+}(S)=\{ma:a\in S\}$ and $E_{•}(S)=\{a^{n-1}:a\in S\}$.

In Sec. 2, as a generalization of Green’s relations, we recall Green’s ∗-relations and obtain some properties. In Sec. 3, we give characterizations of ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}$ (respectively, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}$) as a congruence on semirings and prove that semiring classes determined by these congruences are subvarieties of COS${}_n^{•}$. In Sec. 4, we describe an equivalence condition that ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}$ (respectively, ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}$) is a congruence on a semiring in COS${}_n^{•}$ and show that semiring classes determined by these congruences are also subvarieties of COS${}_n^{•}$.

For other notation and terminology, we use in this paper the reader is referred to [7, 13, 17].

2. Preliminaries

Let $(S,+,\cdot )$ be a semiring. Generally, we denote Green’s relations on the additive (respectively, multiplicative) reduct $(S,+)$ (respectively, $(S,\cdot )$) by ${{ℒ}}^{+}$, ${{ℛ}}^{+}$, ${{𝒟}}^{+}$, ${{\mathscr{H}}}^{+}$ and ${{𝒥}}^{+}$ (respectively, ${{ℒ}}^{•}$, ${{ℛ}}^{•}$, ${{𝒟}}^{•}$, ${{\mathscr{H}}}^{•}$ and ${{𝒥}}^{•}$). We recall equivalence relations defined by Maity and Ghosh [6, Definition 3.5] as follows:

where r and s are the smallest positive integers such that $ra$ and $sb$ are additively regular. They are called Green’s ∗-relations. If $S\in {\mathrm{\text{COS}}}_n^{•}$, then, for every $a\in S$, $2a$ is a additively regular element. So, we have the following.

Definition 2.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

Let $S^{\ast }=\{2a:a\in S\}$. Obviously, $S^{\ast }$ is a completely regular subsemiring of S. Thus, by [11, II, Theorem 3.6], we have the following.

Theorem 2.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

Lemma 2.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then $(E_{+}(S),+,\cdot )$ is a subsemiring of S. In fact, for every $a,b\in S$, we have

Proof.. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. For any $a,b\in S$, take $f={(a+b)}^{n-1}(ma+mb)$. On the one hand, $f={(a+b)}^{n-1}(ma)+{(a+b)}^{n-1}(mb)=m{(a+b)}^{n-1}(a+b)=m(a+b)$. On the other hand, from $ma,mb\in E_{+}(S)$, we have

Thus, $m(a+b)=ma+mb$. □

By Lemma 2.1, we have the following.

Lemma 2.2. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

where k is a positive integer.

Proof. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. For any $a,b\in S$. In the first equality, the claim obviously holds when $k=1$. Assume now that this equality holds for $k-1$ and then, by Lemma 2.1,

since the last term of ${(a+b)}^{k-1}$ is $b^{k-1}$. Similarly, we can prove that the second equality holds. □

Immediately, we have the following results.

Theorem 2.2. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

Proof. Let $f=(2a+mb){((ma+2b)(2a+mb))}^{n-1}$. By Lemma 2.1, $mf=m(a+b)$, since $ma,mb\in E_{+}(S)$. By Lemma 2.2, we have

This shows that $mf=m(a+b+(a+b){(ba+b^2)}^{n-1})$. Therefore, the first equality holds. Note that if we rewrite $f=(2a+mb){(2(m{a}^2+2ba)+m(ab+b^2))}^{n-1}$, then This shows that $mf=m((a+b){(a^2+ba)}^{n-1}+a+b)$. Therefore, the third equation holds. From $f={((ma+2b)(2a+mb))}^{n-1}(ma+2b)$ we can prove that the other equalities hold. □

3. Some of the Subvarieties Associated with ${{\mathscr{H}}}_{\ast }^{+}$

In this section, we give sufficient and necessary conditions that ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}$ and ${{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}$ are congruences on S and prove that the classes of semirings determined by these congruences are subvarieties of COS${}_n^{•}$. By Theorem 2.1, we have

Lemma 3.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

$(\mathrm{\text{i}})$	$a{{\mathscr{H}}}_{\ast }^{+}b\iff (\exists u,v\in S)2a=2u+mv$, $2b=mu+2v$,
$(\mathrm{\text{ii}})$	$a{{ℒ}}_{\ast }^{•}b\iff (\exists u,v\in S)2a=2u{v}^{n-1}{u}^{n-1}$, $2b=2v{u}^{n-1}$,
$(\mathrm{\text{iii}})$	$a{{ℛ}}_{\ast }^{•}b\iff (\exists u,v\in S)2a=2u^{n-1}{v}^{n-1}u$, $2b=2u^{n-1}v$,
$(\mathrm{\text{iv}})$	$a{{𝒟}}_{\ast }^{•}b\iff (\exists u,v\in S)2a=2u{(vu)}^{n-1}$, $2b=2{(vu)}^{n-1}v$,
$(\mathrm{\text{v}})$	$a{{\mathscr{H}}}_{\ast }^{•}b\iff (\exists u,v\in S)2a=2{(vu)}^{n-1}u{(vu)}^{n-1}$, $2b=2{(vu)}^{n-1}v{(vu)}^{n-1}$.

Proof. We only prove (iv), and others can be proved similarly. Let $u=a$, $v=b$. If $a{{𝒟}}_{\ast }^{•}b$, then $4vu=4ba$ in $R_{2b}\cap L_{2a}$ by [5, Proposition 2.3.7]. Hence, by [5, Proposition 2.3.3], $2a=2u{(vu)}^{n-1}$ and $2b=2{(uv)}^{n-1}v$. Conversely, if $u,v\in S$ and let $2a=2u{(vu)}^{n-1}$, $2b=2{(uv)}^{n-1}v$, then $2a=2a{(ba)}^{n-1}$ and $2b=2{(ba)}^{n-1}b$. Therefore, by Theorem 2.1, $2a{{𝒟}}_{\ast }^{•}2b$. □

The conclusions with respect to intersections between above Green’s ∗-relations are given by the following lemma.

Lemma 3.2. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

$(\mathrm{\text{i}})$	$a({{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•})b\iff (\exists u,v\in S)\left\{\begin{array}{cc}2a=2ũ+mṽ,\hfill & ũ=u{v}^{n-1}{u}^{n-1},\hfill \\ 2b=mũ+2ṽ,\hfill & ṽ=v{u}^{n-1},\hfill \end{array}\right.$
$(\mathrm{\text{ii}})$	$a({{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•})b\iff (\exists u,v\in S)\left\{\begin{array}{cc}2a=2ũ+mṽ,\hfill & ũ=u^{n-1}{v}^{n-1}u,\hfill \\ 2b=mũ+2ṽ,\hfill & ṽ=u^{n-1}v,\hfill \end{array}\right.$
$(\mathrm{\text{iii}})$	$a({{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•})b\iff (\exists u,v\in S)\left\{\begin{array}{cc}2a=2ũ+mṽ,\hfill & ũ=u{(vu)}^{n-1},\hfill \\ 2b=mũ+2ṽ,\hfill & ṽ={(vu)}^{n-1}v,\hfill \end{array}\right.$
$(\mathrm{\text{iv}})$	$a({{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•})b\iff (\exists u,v\in S)\left\{\begin{array}{cc}2a=2ũ+mṽ,\hfill & ũ={(vu)}^{n-1}u{(vu)}^{n-1},\hfill \\ 2b=mũ+2ṽ,\hfill & ṽ={(vu)}^{n-1}v{(vu)}^{n-1}.\hfill \end{array}\right.$

Proof. We only prove (iii), and others can be proved similarly. If $a({{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•})b$, then $2a=2a{(ba)}^{n-1}$, $2b=2{(ba)}^{n-1}b$, $ma=mb$ by Theorem 2.1. Let $u=a$, $v=b$. Then

and Conversely, if $2a=2u{(vu)}^{n-1}+m{(vu)}^{n-1}v$, $2b=mu{(vu)}^{n-1}+2{(vu)}^{n-1}v$, then $a{{\mathscr{H}}}_{\ast }^{+}b$. Let $ũ=u{(vu)}^{n-1}$, $ṽ={(vu)}^{n-1}v$. Hence, by Theorem 2.1 and Lemma 3.1, $2ũ=2ũ{(ṽũ)}^{n-1}$ and $2ṽ=2{(ṽũ)}^{n-1}ṽ$. By Lemma 2.2 and Theorem 2.2, we have and Thus $a{{𝒟}}_{\ast }^{•}b$. □

Now, we can establish the following theorem.

Theorem 3.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

$(\mathrm{\text{i}})$	${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill z(2\tilde{x}+mỹ){(z(m\tilde{x}+2ỹ))}^{n-1}& \hfill \approx \hfill & z(2\tilde{x}+mỹ),\hfill \\ \hfill (2z+2\tilde{x}+mỹ){(2z+m\tilde{x}+2ỹ)}^{n-1}& \hfill \approx \hfill & 2z+2\tilde{x}+mỹ,\hfill \\ \hfill (2\tilde{x}+mỹ+2z){(m\tilde{x}+2ỹ+2z)}^{n-1}& \hfill \approx \hfill & 2\tilde{x}+mỹ+2z,\hfill \end{array}$$ where $\tilde{x}=x{y}^{n-1}{x}^{n-1}$, $ỹ=y{x}^{n-1}$.
$(\mathrm{\text{ii}})$	${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill {((m\tilde{x}+2ỹ)z)}^{n-1}(2\tilde{x}+mỹ)z& \hfill \approx \hfill & (2\tilde{x}+mỹ)z,\hfill \\ \hfill {(2z+m\tilde{x}+2ỹ)}^{n-1}(2z+2\tilde{x}+mỹ)& \hfill \approx \hfill & 2z+2\tilde{x}+mỹ,\hfill \\ \hfill {(m\tilde{x}+2ỹ+2z)}^{n-1}(2\tilde{x}+mỹ+2z)& \hfill \approx \hfill & 2\tilde{x}+mỹ+2z,\hfill \end{array}$$ where $\tilde{x}=x^{n-1}{y}^{n-1}x$, $ỹ=x^{n-1}y$.
$(\mathrm{\text{iii}})$	${{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill (2z+2\tilde{x}+mỹ){((2z+m\tilde{x}+2ỹ)(2z+2\tilde{x}+mỹ))}^{n-1}& \hfill \approx \hfill & 2z+2\tilde{x}+mỹ,\hfill \\ \hfill (2\tilde{x}+mỹ+2z){((m\tilde{x}+2ỹ+2z)(2\tilde{x}+mỹ+2z))}^{n-1}& \hfill \approx \hfill & 2\tilde{x}+mỹ+2z,\hfill \end{array}$$ where $\tilde{x}=x{(yx)}^{n-1}$, $ỹ={(yx)}^{n-1}y$.
$(\mathrm{\text{iv}})$	${{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill {(2z+2\tilde{x}+mỹ)}^{n-1}& \hfill \approx \hfill & {(2z+m\tilde{x}+2ỹ)}^{n-1},\hfill \\ \hfill {(2\tilde{x}+mỹ+2z)}^{n-1}& \hfill \approx \hfill & {(m\tilde{x}+2ỹ+2z)}^{n-1},\hfill \end{array}$$ where $\tilde{x}={(yx)}^{n-1}x{(yx)}^{n-1}$, $ỹ={(yx)}^{n-1}y{(yx)}^{n-1}$.

Proof. We only prove (i), and others can be proved similarly. If ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$, then, by Lemma 3.2, $(2x+my)({{\mathscr{H}}}^{+}\wedge {{ℒ}}^{•})(mx+2y)$ where $x=a{b}^{n-1}{a}^{n-1}$, $y=b{a}^{n-1}$ for all $a,b,c\in S$. Hence $c(2x+my){{ℒ}}^{•}c(mx+2y)$, $(2c+2x+my){{ℒ}}^{•}(2c+mx+2y)$ and $(2x+my+2c){{ℒ}}^{•}(mx+2y+2c)$, i.e.

Thus, S satisfies the corresponding identities above.

Conversely, suppose $a({{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•})b$ for any $a,b\in S$. If S satisfies the corresponding identities, then, by Lemma 3.2, there exist $u,v\in S$ such that $2a=2x+my$, $2b=mx+2y$ where $x=u{v}^{n-1}{u}^{n-1}$, $y=v{u}^{n-1}$. Thus, for all $c\in S$,

i.e. $2(ca){(cb)}^{n-1}=2ca$, $(2c+2a){(2c+2b)}^{n-1}=2c+2a$ and $(2a+2c){(2b+2c)}^{n-1}=2a+2c$. Dually, $2(cb){(ca)}^{n-1}=2cb$, $(2c+2b){(2c+2a)}^{n-1}=2c+2b$ and $(2b+2c){(2a+2c)}^{n-1}=2b+2c$. That is to say, $ca{{ℒ}}_{\ast }^{•}cb$, $(c+a){{ℒ}}_{\ast }^{•}(c+b)$ and $(a+c){{ℒ}}_{\ast }^{•}(b+c)$. Hence, ${{ℒ}}_{\ast }^{•}$ is a left congruence on $(S,\cdot )$ and a congruence on $(S,+)$. Since ${{ℒ}}_{\ast }^{•}$ is a right congruence on $(S,\cdot )$, it follows that ${{ℒ}}_{\ast }^{•}$ is a congruence on S. Thus, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$. □

By Theorem 3.1, semiring classes $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ and $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ are varieties. If $n=2$, then this result on $S^{\ast }$ is the same as Theorem 2.3 in [16].

A useful further specialization of this result is provided by the following corollary.

Corollary 3.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

Thus, semiring classes $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}=\mathrm{\Delta }\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}=\mathrm{\Delta }\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}=\mathrm{\Delta }\}$ and $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}=\mathrm{\Delta }\}$ are varieties.

4. Some New Subvarieties Associated with Green’s ∗-relations

In this section, we describe some new subvarieties of COS${}_n^{•}$ by identities. The characterizations for ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}$ and ${{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}$ are given by the following lemma.

Lemma 4.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

Proof. We only prove the third equality. It is obvious that ${{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}\subseteq {{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}$. To show that ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}\subseteq {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$, we need to verify ${{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\subseteq {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$. If $a{{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}b$ for any $a,b\in S$, then there exist $c,d\in S$ such that $a{{𝒟}}_{\ast }^{•}c{{\mathscr{H}}}_{\ast }^{+}d{{𝒟}}_{\ast }^{•}b$. Certainly, $2a=2a{(ca)}^{n-1}{{\mathscr{H}}}^{+}2a{(da)}^{n-1}$. Since ${{𝒟}}^{•}$ is a congruence on $(S,\cdot )$, it follows that $2{(da)}^{n-1}d{{𝒟}}^{•}2{(bc)}^{n-1}b$ by $2a{{𝒟}}^{•}2c$, $2b{{𝒟}}^{•}2d$. Therefore, by Lemma 3.1, $2a{{\mathscr{H}}}^{+}2a{(da)}^{n-1}{{𝒟}}^{•}2{(da)}^{n-1}d{{𝒟}}^{•}2{(bc)}^{n-1}b{{\mathscr{H}}}^{+}2b{(db)}^{n-1}=2b$, i.e. $a{{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}b$. This implies that ${{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\subseteq {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$. So, it is easy to check that ${{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$ is an equivalence containing ${{\mathscr{H}}}_{\ast }^{+}$ and ${{𝒟}}_{\ast }^{•}$. Thus, ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}\subseteq {{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$ and so ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}={{\mathscr{H}}}_{\ast }^{+}\circ {{𝒟}}_{\ast }^{•}\circ {{\mathscr{H}}}_{\ast }^{+}$. Other equalities can be proved similarly. □

Further characterizations are available.

Lemma 4.2. Let $S\in {\mathrm{\text{COS}}}_n^{•}$ and $a,b\in S$. Then

$(\mathrm{\text{i}})$	$a({{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•})b\iff a{{\mathscr{H}}}_{\ast }^{+}a{b}^{n-1}$, $b{{\mathscr{H}}}_{\ast }^{+}b{a}^{n-1}$,
$(\mathrm{\text{ii}})$	$a({{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•})b\iff a{{\mathscr{H}}}_{\ast }^{+}{b}^{n-1}a$, $b{{\mathscr{H}}}_{\ast }^{+}{a}^{n-1}b$,
$(\mathrm{\text{iii}})$	$a({{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•})b\iff a{{\mathscr{H}}}_{\ast }^{+}a{(ba)}^{n-1}$, $b{{\mathscr{H}}}_{\ast }^{+}{(ba)}^{n-1}b$,
$(\mathrm{\text{iv}})$	$a({{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•})b\iff a^{n-1}{{\mathscr{H}}}_{\ast }^{+}{b}^{n-1}$.

Proof. We only prove (iii) and others can be proved similarly. If $a({{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•})b$ for any $a,b\in S$, then there exist $c,d\in S$ such that $a{{\mathscr{H}}}_{\ast }^{+}c{{𝒟}}_{\ast }^{•}d{{\mathscr{H}}}_{\ast }^{+}b$ by Lemma 4.1. So, by Theorem 2.1, $ma=mc=mc{(dc)}^{n-1}=mc{((md)(mc))}^{n-1}=ma{(ba)}^{n-1}$. Thus, $a{{\mathscr{H}}}_{\ast }^{+}a{(ba)}^{n-1}$. Similarly, $b{{\mathscr{H}}}_{\ast }^{+}{(ba)}^{n-1}b$. Conversely, if $a{{\mathscr{H}}}_{\ast }^{+}a{(ba)}^{n-1}$ and $b{{\mathscr{H}}}_{\ast }^{+}{(ba)}^{n-1}b$, then, by Lemma 3.1, $a{{\mathscr{H}}}_{\ast }^{+}a{(ba)}^{n-1}{{𝒟}}_{\ast }^{•}{(ba)}^{n-1}b{{\mathscr{H}}}_{\ast }^{+}b$. Thus, by Lemma 4.1, $a({{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•})b$. □

Lemma 4.3. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

The characterization of ${{ℒ}}_{\ast }^{•}$ (respectively, ${{ℛ}}_{\ast }^{•}$, ${{𝒟}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{•}$) is given by the following lemma.

Lemma 4.4. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

$(\mathrm{\text{i}})$	${{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill 2zx{y}^{n-1}{x}^{n-1}{(zy{x}^{n-1})}^{n-1}& \hfill \approx \hfill & 2zx{y}^{n-1}{x}^{n-1},\hfill \\ \hfill 2(z+x{y}^{n-1}{x}^{n-1}){(z+y{x}^{n-1})}^{n-1}& \hfill \approx \hfill & 2z+2x{y}^{n-1}{x}^{n-1},\hfill \\ \hfill 2(x{y}^{n-1}{x}^{n-1}+z){(y{x}^{n-1}+z)}^{n-1}& \hfill \approx \hfill & 2x{y}^{n-1}{x}^{n-1}+2z.\hfill \end{array}$$
$(\mathrm{\text{ii}})$	${{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill 2{(x^{n-1}yz)}^{n-1}{x}^{n-1}{y}^{n-1}xz& \hfill \approx \hfill & 2x^{n-1}{y}^{n-1}xz,\hfill \\ \hfill 2{(z+x^{n-1}y)}^{n-1}(z+x^{n-1}{y}^{n-1}x)& \hfill \approx \hfill & 2z+2x^{n-1}{y}^{n-1}x,\hfill \\ \hfill 2{(x^{n-1}y+z)}^{n-1}(x^{n-1}{y}^{n-1}x+z)& \hfill \approx \hfill & 2x^{n-1}{y}^{n-1}x+2z.\hfill \end{array}$$
$(\mathrm{\text{iii}})$	${{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill 2(z+x{(yx)}^{n-1}){((z+{(yx)}^{n-1}y)(z+x{(yx)}^{n-1}))}^{n-1}& \hfill \approx \hfill & 2z+2x{(yx)}^{n-1},\hfill \\ \hfill 2(x{(yx)}^{n-1}+z){(({(yx)}^{n-1}y+z)(x{(yx)}^{n-1}+z))}^{n-1}& \hfill \approx \hfill & 2x{(yx)}^{n-1}+2z.\hfill \end{array}$$
$(\mathrm{\text{iv}})$	${{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill {(2z+2{(yx)}^{n-1}x{(yx)}^{n-1})}^{n-1}& \hfill \approx \hfill & {(2z+2{(yx)}^{n-1}y{(yx)}^{n-1})}^{n-1},\hfill \\ \hfill {(2{(yx)}^{n-1}x{(yx)}^{n-1}+2z)}^{n-1}& \hfill \approx \hfill & {(2{(yx)}^{n-1}y{(yx)}^{n-1}+2z)}^{n-1}.\hfill \end{array}$$

Proof. We only prove (i) and others can be proved similarly. By Lemma 3.1, $2a{b}^{n-1}{a}^{n-1}{{ℒ}}^{•}2b{a}^{n-1}$ for all $a,b\in S$. If ${{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$, then

for all $c\in S$, i.e. Thus, S satisfies the corresponding identities above.

Conversely, let $a{{ℒ}}_{\ast }^{•}b$ for any $a,b\in S$. If S satisfies the corresponding identities, then there exist $u,v\in S$ such that $a=u{v}^{n-1}{u}^{n-1}$ and $b=v{u}^{n-1}$ by Lemma 3.1. Also, for all $c\in S$,

i.e. $2(ca){(cb)}^{n-1}=2ca$, $2(c+a){(c+b)}^{n-1}=2(c+a)$ and $2(a+c){(b+c)}^{n-1}=2(a+c)$. Dually, $2(cb){(ca)}^{n-1}=2cb$, $2(c+b){(c+a)}^{n-1}=2(c+b)$ and $2(b+c){(a+c)}^{n-1}=2(b+c)$. That is to say, $ca{{ℒ}}_{\ast }^{•}cb$, $(c+a){{ℒ}}_{\ast }^{•}(c+b)$ and $(a+c){{ℒ}}_{\ast }^{•}(b+c)$. Thus, ${{ℒ}}_{\ast }^{•}$ is a left congruence on $(S,\cdot )$ and a congruence on $(S,+)$. But ${{ℒ}}_{\ast }^{•}$ is a right congruence on $(S,\cdot )$, it follows that ${{ℒ}}_{\ast }^{•}$ is a congruence on S. □

By Lemma 4.4, semiring classes $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ and $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ are varieties.

By Lemmas 4.3 and 4.4, we obtain the following theorem.

Theorem 4.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

$(\mathrm{\text{i}})$	${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill mzx{y}^{n-1}{x}^{n-1}{(zy{x}^{n-1})}^{n-1}& \hfill \approx \hfill & mzx{y}^{n-1}{x}^{n-1},\hfill \\ \hfill m(z+x{y}^{n-1}{x}^{n-1}){(z+y{x}^{n-1})}^{n-1}& \hfill \approx \hfill & mz+mx{y}^{n-1}{x}^{n-1},\hfill \\ \hfill m(x{y}^{n-1}{x}^{n-1}+z){(y{x}^{n-1}+z)}^{n-1}& \hfill \approx \hfill & mx{y}^{n-1}{x}^{n-1}+mz.\hfill \end{array}$$
$(\mathrm{\text{ii}})$	${{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill m{(x^{n-1}yz)}^{n-1}{x}^{n-1}{y}^{n-1}xz& \hfill \approx \hfill & m{x}^{n-1}{y}^{n-1}xz,\hfill \\ \hfill m{(z+x^{n-1}y)}^{n-1}(z+x^{n-1}{y}^{n-1}x)& \hfill \approx \hfill & mz+m{x}^{n-1}{y}^{n-1}x,\hfill \\ \hfill m{(x^{n-1}y+z)}^{n-1}(x^{n-1}{y}^{n-1}x+z)& \hfill \approx \hfill & m{x}^{n-1}{y}^{n-1}x+mz.\hfill \end{array}$$
$(\mathrm{\text{iii}})$	${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill m(z+x{(yx)}^{n-1}){((z+{(yx)}^{n-1}y)(z+x{(yx)}^{n-1}))}^{n-1}& \hfill \approx \hfill & mz+mx{(yx)}^{n-1},\hfill \\ \hfill m(x{(yx)}^{n-1}+z){(({(yx)}^{n-1}y+z)(x{(yx)}^{n-1}+z))}^{n-1}& \hfill \approx \hfill & mx{(yx)}^{n-1}+mz.\hfill \end{array}$$
$(\mathrm{\text{iv}})$	${{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$ if and only if S satisfies the following identities: $$\begin{array}{ccc}\hfill m{(z+{(yx)}^{n-1}x{(yx)}^{n-1})}^{n-1}& \hfill \approx \hfill & m{(z+{(yx)}^{n-1}y{(yx)}^{n-1})}^{n-1},\hfill \\ \hfill m{({(yx)}^{n-1}x{(yx)}^{n-1}+z)}^{n-1}& \hfill \approx \hfill & m{({(yx)}^{n-1}y{(yx)}^{n-1}+z)}^{n-1}.\hfill \end{array}$$

Proof. We only prove (i) and others can be proved similarly. If ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$, then, by Lemma 4.3

for all $a,b,c\in S$, i.e. Thus, S satisfies the corresponding identities above.

Conversely, if S satisfies the corresponding identities, then

for all $a,b,c\in S$. Hence, by Lemma 4.3, ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)$. □

From Theorem 4.1, semiring classes $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ and $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}\in \mathrm{\text{Con}}(S)\}$ are varieties.

We immediately have the following corollary.

Corollary 4.1. Let $S\in {\mathrm{\text{COS}}}_n^{•}$. Then

Semiring classes $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}=\nabla \}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}=\nabla \}$, $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}=\nabla \}$ and $\{S\in {\mathrm{\text{COS}}}_n^{•}:{{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}=\nabla \}$ are varieties.

ORCID

Yilin Mao  https://orcid.org/0009-0007-2781-2817

Yong Shao  https://orcid.org/0000-0003-3365-8987

Yanliang Cheng  https://orcid.org/0009-0003-6862-9446