Nil extensions of Clifford ordered semigroup

TL;DR

This paper characterizes all ordered semigroups that are nil extensions of various classes such as Clifford, left Clifford, group-like, and left group-like semigroups, providing a comprehensive classification.

Contribution

It provides a complete description of ordered semigroups that are nil extensions of specific well-known classes, expanding the understanding of their structure.

Findings

Characterization of nil extensions of Clifford ordered semigroups

Description of nil extensions of left Clifford ordered semigroups

Classification of nil extensions of group-like and left group-like ordered semigroups

Abstract

In this paper we describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups.

Equations

$$a\le a(xa{)}^n,\text{for all}n\in \mathbb{N}\text{and some }x\in S.$$

$$(xa{)}^{m_1}\le (xa{)}^{m_1}{t}_1(xa{)}^{m_1}sa,\text{by the first condition}.$$

$$as\le as{t}_2(ax{)}^{m_2}\text{for some}{t}_2\in S.$$

$$t\le t(zt{)}^n\text{for all}n\in \mathbb{N}\text{for some}z\in T.$$

$$(xb{)}^{m_4}\le (xb{)}^{m_4}{t}_4(xb{)}^{m_4}a\text{for some }{m}_4\in \mathbb{N}\text{and}{t}_4\in S.$$

$$a$$

$$=b{t}_{5}a;\text{where}{t}_5=(xb{)}^{m_4}{t}_4(xb{)}^{m_4}.$$

$$a$$

$$=a{t}_{7}b(xb{)}^{m_4}{t}_4(xb{)}^{m_4}$$

$$\le a{t}_{7}b(xb{)}^{m_4}{t}_4(xb{)}^{m_4}{t}_4(xb{)}^{m_4}a,\text{by}(\text{ref}eq5.3).$$

$$bs\le bybs\le (by{)}^{n}bs\text{for}y\in S.$$

$$bs$$

$$\le bstbs;\text{where}t=b{s}^{m-1}zb{s}^m.$$

$$a$$

$$=ata,\text{where}t=a^{m-1}{z}_1{a}^{m}b{a}^{n}z{a}^{n-1}.$$

$$a\le (ax{)}^{n}a\text{for all}n\in \mathbb{N},$$

$$x{a}^{m}y\le x{a}^{m}y{z}_{1}x{a}^{m}y.$$

$$z_{1}x{a}^{m}y\le (a^{m}y)u_1(z_{1}x),\text{for some}{u}_1\in K,\text{since}S\text{Clifford}.$$

$$a^m(y{u}_1{z}_1)\le (y{u}_1{z}_1)u_2{a}^m.$$

$$x{a}^{m}y$$

$$\le x{a}^{m}y{a}^m(y{u}_1{z}_1)x,\text{by (refeq5.8)}$$

$$\le x{a}^m{y}^2{u}_1{z}_1{u}_2{a}^{m}x,\text{by (refeq5.9)}.$$

$$x{a}^{m}y\le x{a}^{m}y{z}_{1}x{a}^m{y}^2{u}_1{z}_1{u}_2{a}^{m}x.$$

$$x{a}^{m}y$$

$$\le x{a}^{m}ysy{a}^{m}x;\text{where}s=z_{1}x{u}_1{z}_1{u}_2{u}_3{a}^{m}y.$$

$$xs$$

$$\le xt(xt{)}^{n-1}xs.$$

$$xs$$

$$\le xsqxs;\text{where}q=(xt{)}^{m_2}xt{s}_{2}xt(xt{)}^{m_2}.$$

$$ab$$

$$\le (abra)(bra{)}^{n-1}b,\text{for all}n\in \mathbb{N}.$$

$$abra(bra{)}^{m_3}b\le b(bra{)}^{m_3}abra{p}_{1}abra(bra{)}^{m_3}b\text{for some}{p}_1\in S\text{and}{m}_3\in \mathbb{N}.$$

$$ab$$

$$\le b(bra{)}^{m_3}abra{p}_{1}abra(bra{)}^{m_3-1}brab$$

$$\le bgab;\text{where}g=(bra{)}^{m_3}abra{p}_{1}abra(bra{)}^{m_3-1}br\in T.$$

$$ab\le b(gabra(bra{)}^{m_4}b{p}_{2}b(bra{)}^{m_4})abra=bua;\text{where}u=gabra(bra{)}^{m_4}b{p}_{2}b(bra{)}^{m_4}abr\in T.$$

$$a\le zabtab\text{for some}t\in S.$$

Full text

Key Words and phrases: nil extension, regular ordered semigroups, group like, left group like, Clifford and left Clifford ordered semigroups.

2010 Mathematics subject Classification: 20M10; 06F05.

1 Introduction:

Nil extensions of a semigroup (without order), are precisely the ideal extensions by a nil semigroup. In 1984, S. Bogdanovi$\acute{c}$ and S. Mili$\acute{c}$ [2] characterized the semigroups (without order) which are nil extensions of completely simple semigroups, where as, a similar work was done by J. L Galbiati and M.L Veronesi [14] in 1980. Authors like S. Bogdanovi$\acute{c}$, M. Ciri$\acute{c}$, Beograd have investigated this type extensions for regular semigroup, group, periodic semigroup as well as completely regular semigroup(see [4], [5]).

The notion of ideal extensions in ordered semigroups is actually introduced by N. Kehayopulu and M. Tsingelis [8]. In [11], they have worked on ordered semigroups which are nil extensions of Archimedean ordered semigroups. The concepts of nil extensions have been extended to ordered semigroups by Y. Cao [7], with characterizing ordered semigroups which are nil extensions of simple, left simple, t-simple ordered semigroups. Further he described complete semilattices of nil extensions of such ordered semigroups.

The aim of this paper is to describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups. This research originates from the research papers [4], [5].

Our paper organized as follows. The basic definitions and properties of ordered semigroups are presented in Section 2. Section 3 is devoted to characterizing the nil extensions of Clifford and left Clifford ordered semigroups.

2 Preliminaries:

In this paper $\mathbb{N}$ will provide the set of all natural numbers. An ordered semigroup is a partiality ordered set $S$, and at the same time a semigroup $(S,\cdot)$ such that $(\textrm{forall}\;a,\;b,\;x\in S)\;a\leq b\Rightarrow\;xa\leq xb\;\textrm{and}\;ax\leq bx$. It is denoted by $(S,\cdot,\leq)$. For an ordered semigroup $S$ and $H\subseteq S$, denote $(H]:=\{t\in S:t\leq h,\;\textrm{for some}\;h\in H\}$. An element $z$ in $S$ is called zero of $S$ if $z<x$ and $zx=xz=z$ for every $x\in S$.

Let $I$ be a nonempty subset of an ordered semigroup $S$. $I$ is a left (right) ideal of $S$, if $SI\subseteq I\;(IS\subseteq I)$ and $(I]=I$. $I$ is an ideal of $S$ if $I$ is both a left and a right ideal of $S$. An (left, right) ideal $I\;ofS$ is proper if $I\neq S$. The intersection of all ideals of an ordered semigroup $S$, if nonempty, is called the kernel of $S$ and is denoted by $K(S)$.

An ordered semigroup $S$ is called a group like ordered semigroup if for all $a,b\in S\;\textrm{there are}\;x,y\in S\;\textrm{such \;that}\;a\leq xb\;\textrm{and}\;a\leq by$ [1] and $S$ left group like if for all $a,b\in S\;\textrm{there is}\;x\in S$ such that $a\leq xb$ [1]. Kehayopulu [9] defined Greens relations on an ordered semigroup $S$ as follows: $a\mathcal{L}b\;if\;L(a)=L(b),\;a\mathcal{R}b\;\textrm{if}\;R(a)=R(b),\;a\mathcal{J}b\;\textrm{if}\;I(a)=I(b),\;\textrm{and}\;\mathcal{H}=\;\mathcal{L}\cap\;\mathcal{R}$. These four relations $\mathcal{L},\;\mathcal{R},\;\mathcal{J}\;\textrm{and}\;\mathcal{H}$ are equivalence relations. In an ordered semigroup $S$, an equivalence relation $\rho$ is called left (right) congruence if for $a,b,c\in S\;a\;\rho\;b\;\textrm{implies}\;ca\;\rho\;cb\;(ac\;\rho\;bc)$. $\rho$ is congruence if it is both left and right congruence. A congruence $\rho$ on $S$ is called semilattice if for all $a,b\in S\;a\;\rho\;a^{2}\;\textrm{and}\;ab\;\rho\;ba$. A semilattice congruence $\rho$ on $S$ is called complete if $a\leq b$ implies $(a,ab)\in\rho$. The ordered semigroup $S$ is called complete semilattice of subsemigroup of type $\tau$ if there exists a complete semilattice congruence $\rho$ such that $(x)_{\rho}$ is a type $\tau$ subsemigroup of $S$. Equivalently: There exists a semilattice $Y$ and a family of subsemigroups $\{S\}_{\alpha\in Y}$ of type $\tau$ of $S$ such that:

$S_{\alpha}\cap S_{\beta}=\;\phi$ for any $\alpha,\;\beta\in\;Y\;\textrm{with}\;\alpha\neq\beta,$ 2. 2.

$S=\bigcup_{\alpha\;\in\;Y}\;S_{\alpha},$ 3. 3.

$S_{\alpha}S_{\beta}\;\subseteq\;S_{\alpha\;\beta}$ for any $\alpha,\;\beta\in\;Y,$ 4. 4.

$S_{\beta}\cap(S_{\alpha}]\neq\phi$ implies $\beta\;\preceq\;\alpha,$ where $\preceq$ is the order of the semilattice $Y$ defined by

$\preceq:=\{(\alpha,\;\beta)\;\mid\;\alpha=\alpha\;\beta\;(\beta\;\alpha)\}$ [10].

Let $S$ be an ordered semigroup with the zero [math]. An element $a\in S$ is called a nilpotent if $a^{n}=0$ for some $n\in\mathbb{N}$. The set of all nilpotents of $S$ is denoted by $Nil(S)$. $S$ is called nil ordered semigroup (nilpotent) [7] if $S=Nil(S)$.

Due to Cao [7] the definition of nil extension of ordered semigroup is as follows.

Definition 2.1**.**

[7]** Let $I$ be an ideal of an ordered semigroup $S$. Then $(S/I,\cdot,\preceq)$ is called the Rees factor ordered semigroup of $S\;modulo\;I$, and $S$ is called an ideal extension of $I$ by ordered semigroup $S/I$. An ideal extension $S\;of\;I$ is called a nil-extension of $I$ if $S/I$ is a nil ordered semigroup.

Lemma 2.2**.**

[7]** Let $S$ be an ordered semigroup and $I$ an ideal of $S$. Then the following are equivalent:

• (i)

$S$* is a nil-extension of $I$;*

• (ii)

$(\forall a\in S)(\exists m\in\mathbb{N})\;a^{m}\in I$.

In [1], we have introduced the notion of Clifford and left Clifford ordered semigroups and characterized their structural representation. For the correspondences of the results of this paper we are stating some of them.

Definition 2.3**.**

[1]** A regular ordered semigroup $S$ is called Clifford ordered (left Clifford ordered ) semigroup if for all $a,b\in S$ and $ab\in(bSa]\;(ab\in(Sa])$.

Theorem 2.4**.**

[1]** Let $S$ be a regular ordered semigroup. Then followings hold in $S$:

$S$* is Clifford if and only if $\mathcal{L}=\mathcal{R}$.* 2. 2.

$\mathcal{L}$* is a complete semilattice congruence if $S$ is Clifford.* 3. 3.

$S$* is Clifford ordered semigroup if and only if it is a complete semilattice of group like ordered semigroups.*

Theorem 2.5**.**

[1]** Let $S$ be a regular ordered semigroup. Then followings hold in $S$:

$\mathcal{L}$* is a complete semilattice congruence if $S$ is left Clifford.* 2. 2.

$S$* is left Clifford ordered semigroup if and only if it is a complete semilattice of left group like ordered semigroups.*

3 Main Results:

Now we describe all those ordered semigroups which are the nil extensions of Clifford, left Clifford, group like, left group like ordered semigroups. We omit the proof of the following lemma as it is straightforward.

Lemma 3.1**.**

An ordered semigroup $S$ is left group like ordered semigroup if and only if $a\in(aSab]$ if for all $a,\;b\in S$.

Theorem 3.2**.**

An ordered semigroup $S$ is a nil extension of a left group like ordered semigroup if and only if for every $a,b\in S,\;there\;exists\;n\in\mathbb{N}\;such\;that\;a^{n}\in(a^{n}Sa^{n}b]$ and for every $a\in S,\;b\in Reg_{\leq}(S),\;a\leq ba$ implies $a\leq axb\;for\;some\;x\in S$.

Proof.

Suppose that $S$ is a nil extension of a left group like ordered semigroup $K$ and $a,b\in S$. Then there is $m\in\mathbb{N}$ such that $a^{m}\in K$. Regularity of $K$ implies that $a^{m}\leq a^{m}xa^{m}$ for some $x\in K$. Further, for $xa^{m},a^{m}b\in K$; the left simplicity of $K$ yields that $xa^{m}\leq ya^{m}b,\;\textrm{for some}\;y\in K$. This gives that $a^{m}\leq a^{m}xa^{m}\leq a^{m}ya^{m}b$. Next let $b\in Reg_{\leq}(S)$ and $a\in S$ such that $a\leq ba$. Since $b\in Reg_{\leq}(S),\;\textrm{there exists}\;z\in S$ such that $b\leq b(zb)^{n}\;\textrm{for all}\;n\in\mathbb{N}$. Then for some $n_{1}\in\mathbb{N}$, $(zb)^{n_{1}}\in K$; whence $b(zb)^{n_{1}}\in K$. Thus $b\in K$ and so $ba\in K$. Thus $a\in K$. Since $K$ is a left group like ordered semigroup, for $a,ba\in K$ it follows that $a\leq asba$ for some $s\in K$, by Lemma 3.1. Thus the given conditions follow.

Conversely, assume that given conditions hold in $S$. Let $a\in S$ be arbitrary. Then by given condition we have $a^{m}\leq a^{m}xa^{m+1}$, for some $x\in S$ and $m\in\mathbb{N}$. This implies $a^{m+1}\in Reg_{\leq}(S)$ and so $Reg_{\leq}(S)\neq\phi$. Denote $T=Reg_{\leq}(S)$.

Let us choose $s\in S\;\textrm{and}\;a\in T$. Then the definition of $T$ implies

[TABLE]

Thus $sa\leq sa(xa)^{n},\;\textrm{for all}\;n\in\mathbb{N}$. Now for $xa,sa\in S,\;\textrm{there exists}\;m_{1}\in\mathbb{N}$ and $t_{1}\in S$ such that

[TABLE]

Then $a\leq a(xa)^{m_{1}}$ implies $sa\leq sa(xa)^{m_{1}}$ and hence $sa\leq sa(xa)^{m_{1}}t_{1}(xa)^{m_{1}}sa$; where $(xa)^{m_{1}}t_{1}(xa)^{m_{1}}\in S$. So $sa\in T$. Also $a\leq(ax)^{n}a\;\textrm{for all}\;n\in\mathbb{N}$. Let $m_{2}\in\mathbb{N}$ be such that $(ax)^{m_{2}}\in T$. Then $as\leq(ax)^{m_{2}}as$ implies by the second condition that

[TABLE]

Denote $t=(ax)^{m_{2}}$. Then the definition of $T$ implies

[TABLE]

Then using the first condition for $zt,as\in S$ we have that $(zt)^{m_{3}}\leq(zt)^{m_{3}}t_{3}(zt)^{m_{3}}as\\ \;\textrm{for some}\;m_{3}\in\mathbb{N}\;\textrm{and}\;t_{3}\in S$. That is $t\leq t(zt)^{m_{3}}t_{3}(zt)^{m_{3}}as$, by (2). So from (1) we have $as\leq ast_{2}t(zt)^{m_{3}}t_{3}(zt)^{m_{3}}as$, and hence $as\in T$.

Next choose $a\in S\;\textrm{and}\;b\in T$ such that $a\leq b$. Since $b\in T$ there is $x\in S$ such that $b\leq b(xb)^{n}$ for all $n\in\mathbb{N}$. Now for $xb,a\in S$, it follows from the first condition that

[TABLE]

So $a\leq b$ implies that

[TABLE]

Since $b\in T$, by above we have $\;bt_{5}\in T$. Say $bt_{5}=t_{6}$. Then for $a\in S,\;\textrm{and}\;t_{6}\in T,\;a\leq t_{6}a$ yields that $a\leq at_{7}t_{6}$ for some $t_{7}\in S$, by second condition. Therefore

[TABLE]

Clearly $t_{7}b(xb)^{m_{4}}t_{4}(xb)^{m_{4}}t_{4}(xb)^{m_{4}}\in T$, as $b\in T$. Thus $a\in T$, which shows that $T$ is an ideal of $S$.

Finally let $c,d\in T$. Then there is $v\in S\;\textrm{such that}\;c\leq c(vc)^{n}\;\textrm{for all}\;n\in\mathbb{N}$. Now for $vc,d\in S,\;\textrm{there exists}\;t_{8}\in S$ such that $c\leq c(vc)^{m_{5}}t_{8}(vc)^{m_{5}}d\;\textrm{for some}\;m_{5}\in\mathbb{N}$. Since $c\in T,\;c(vc)^{m_{5}}t_{8}(vc)^{m_{5}}\in T$. Hence $T$ is left simple. Thus $T$ is left group like ordered semigroup such that for every $a\in S\;\textrm{there is}\;m\in\mathbb{N},\;a^{m}\in T$. Hence $S$ is nil extension of a left group like ordered semigroup $T$. ∎

In the following result we provide an independent proof of Corollary 5.2 of [7].

Theorem 3.3**.**

An ordered semigroup $S$ is a nil extension of a group like ordered semigroup if and only if for all $a,b\in S,\;there\;exists\;n\in\mathbb{N}\;such\;that\;a^{n}\in(b^{n}Sb^{n}]$.

Proof.

Suppose that $S$ is a nil extension of a group like ordered semigroup $G$ and $a,b\in S$. Then there exists $n\in\mathbb{N}\;\textrm{such that}\;a^{n},b^{n}\in G$. Since $G$ is a group like ordered semigroup, there exists $u\in G\;\textrm{such that}\;a^{n}\leq b^{n}u$. Also for $u,b^{n}\in G\;\textrm{there exists}\;x\in G\;\textrm{such that}\;u\leq xb^{n}$. This implies $b^{n}u\leq b^{n}xb^{n}$. Thus $a^{n}\leq b^{n}xb^{n}\;\textrm{and hence}\;a^{n}\in(b^{n}Sb^{n}]$.

Conversely, let us assume that given condition holds in $S$. Choose $a\in S$. Then for some $m\in\mathbb{N}\;\textrm{and}\;x\in S,\;a^{m}\leq a^{m}xa^{m}$. Thus $Reg_{\leq}(S)\neq\phi$. Say $G=Reg_{\leq}(S)$. So for every $a\in S,\;\textrm{there exists}\;m\in\mathbb{N}$ such that $a^{m}\in G$. Let us consider $b\in G\;\textrm{and}\;s\in S$. Then for all $n\in\mathbb{N}$

[TABLE]

Using the given condition for $bs,by\in S$, we obtain $(by)^{m}bs\leq(bs)^{m}z(bs)^{m+1}\;\textrm{for}\;z\in S\;\textrm{and}\;m\in\mathbb{N}$. This yields that

[TABLE]

Thus $bs\in G$. Similarly $sb\in G$.

Next let $a\in S$ and $b\in G$ be such that $a\leq b$. Since $b\in G$ there exists $x\in S$ such that $b\leq bxb$ and hence $b\leq(bx)^{n}b(xb)^{n}$ for all $n\in\mathbb{N}$, which implies that

[TABLE]

So $a\in G$. Hence $G$ is an ideal of $S$.

Finally, consider $a,b\in G$. Then there exists $x\in S$ such that

[TABLE]

and so by the given condition it follows that $a\leq b^{m^{\prime}}z^{\prime}b^{m^{\prime}}a$ for some $m^{\prime}\in\mathbb{N}\;\textrm{and}\;z^{\prime}\in S$. This gives that $a\leq bu\;\textrm{for some}\;u=b^{m^{\prime}-1}z^{\prime}b^{m^{\prime}}a\in G$. Similarly there is some $v\in S$ such that $a\leq vb$. This shows that $G$ is a group like ordered semigroup. Hence $S$ is a nil extension of a group like ordered semigroup $G$. ∎

Theorem 3.4**.**

An ordered semigroup $S$ is a nil extension of a Clifford ordered semigroup if and only if for every $x,a,y\in S,\;there\;exists\;n\in\mathbb{N}\;such\;that\;xa^{n}y\in(xa^{n}ySya^{n}x]\cap(ya^{n}xSxa^{n}y]$ and $a\in S,\;b\in Reg_{\leq}(S)$ such that $a\leq b$, implies $a\in(Sab]$.

Proof.

First suppose that $S$ is a nil extension of a Clifford ordered semigroup $K$. Let $x,a,y\in S$. Then there is $\;m\in\mathbb{N}$ such that $a^{m}\in K$. Since $K$ is an ideal of $S$, $xa^{m}y\in K$. Since $K$ is a regular, there exists $z_{1}\in K$ such that

[TABLE]

Now $z_{1}x,a^{m}y\in K$ implies that

[TABLE]

Similarly for $a^{m},(yu_{1}z_{1})\in K$ there is $u_{2}\in S$ such that

[TABLE]

Therefore

[TABLE]

Thus

[TABLE]

Also, for $a^{m}y^{2},u_{1}z_{1}u_{2}\in K\;\textrm{there exists}\;u_{3}\in K$ such that $a^{m}y^{2}u_{1}z_{1}u_{2}\leq u_{1}z_{1}u_{2}u_{3}a^{m}y^{2}$. Then from (8), we obtain that

[TABLE]

Therefore $xa^{m}y\in(xa^{m}ySya^{m}x]$. Similarly $xa^{m}y\in(ya^{m}xSxa^{m}y]$.

Now $K\subseteq Reg_{\leq}(S)$ implies that $Reg_{\leq}(S)\neq\phi$. Consider $b\in Reg_{\leq}(S)\;\textrm{and}\;a\in S\;\textrm{such that}\;a\leq b$. Since $b\in Reg_{\leq}(S),\;\textrm{there exists}\;z\in S$ such that $b\leq(bz)^{n}b\;\textrm{for all}\;n\in\mathbb{N}$. Since $S$ is a nil extension of $K,\;\textrm{there exists}\;n_{1}\in\mathbb{N}$ such that $(bz)^{n_{1}}\in K$. This gives $(bz)^{n_{1}}b\in K$, which gives $b\in K$ and so $a\in K$, since $K$ is an ideal of $S$. Since $K$ is a Clifford ordered semigroup, by Theorem 2.4(ii) $\mathcal{L}$ is a congruence on $S$. Since $a,b\in K$ we have $\;a\mathcal{L}ab$ and hence $a\subseteq(Sab]$.

Conversely, let us assume that given conditions hold in $S$. Let $a\in S$ be arbitrary. Then by the first condition there exists $n\in\mathbb{N}$ such that $a^{n+2}\leq a^{n+2}xa^{n+2},\\ \;\textrm{for some}\;x\in S$. Thus $Reg_{\leq}(S)\neq\phi$. Say $T=Reg_{\leq}(S)$. It is now clear that for each $a\in S,\;\textrm{there exists }\;m\in\mathbb{N}$ such that $a^{m}\in T$.

Let $s\in S$ and $x\in T$. Then for all $n\in\mathbb{N}$ and for some $\;t\in S,\;x\leq(xt)^{n}x$ which implies that $sx\leq sx(tx)^{n-1}tx,\;\textrm{for all}\;n\in\mathbb{N}$. By first condition there are $s_{1}\in S\;\textrm{and}\;m_{1}\in\mathbb{N}$ such that $sx\leq sx(tx)^{m_{1}}txs_{1}tx(tx)^{m_{1}}sx$ and thus $sx\leq sxpsx;\;\textrm{where}\;p=(tx)^{m_{1}}txs_{1}tx(tx)^{m_{1}}$. Also for every $n\in\mathbb{N}$,

[TABLE]

So there is $m_{2}\in\mathbb{N}$ such that

[TABLE]

Thus $sx,sx\in T$.

To show $T$, a Clifford ordered semigroup, choose $a,b\in T$. Then there is $r\in S$ such that

[TABLE]

Now for $abra,bra,b\in S$, the first condition yields that

[TABLE]

Therefore from (3)

[TABLE]

Similarly there are $m_{4}\in\mathbb{N}$ and $p_{2}\in S$ such that $ab\leq abra(bra)^{m_{4}}bp_{2}b(bra)^{m_{4}}abra$. So from (3),

[TABLE]

Hence $T$ is Clifford ordered semigroup.

Now let $a\leq b$ for some $a\in S$ and $b\in T$. Then by the second condition, there is $z\in S$ such that $a\leq zab,\;\textrm{that is}$,

[TABLE]

Since $T$ is Clifford ordered semigroup, for $zabt,ab\in T$ it follows that

[TABLE]

Similarly for $bp_{3}za,bt\in T$, we have $bp_{3}zabt\leq btp_{4}bp_{3}za\;\textrm{for some}\;p_{4}\in S$.

The last two inequalities together with (11) yields that $a\leq aha,\;\textrm{where}\;h=btp_{4}bp_{3}z$. Thus $a\in T$ and so $T$ is an ideal of $S$. Hence $S$ is a nil extension of a Clifford ordered semigroup $T$. ∎

Theorem 3.5**.**

An ordered semigroup $S$ is a nil extension of a left Clifford ordered semigroup if and only if for every $x,a,y\in S,\;there\;exists\;n\in\mathbb{N}\;such\;that\;xa^{n}y\in(xa^{n}ySya^{n}x]\cap(xa^{n}ySxa^{n}y]$ and $a\in S,\;b\in Reg_{\leq}(S)$ such that $a\leq b$ implies $a\leq azab\;for\;some\;z\in S$.

Proof.

Let $S$ be a nil extension of a left Clifford ordered semigroup $K$. Choose $x,a,y\in S$. Then there exists $\;m\in\mathbb{N}$ such that $a^{m}\in K$. Since $K$ is an ideal of $S$, $xa^{m}y\in K$. Also the regularity of $K$ yields that

[TABLE]

Since $K$ is a left Clifford ordered semigroup and $z_{1}x,a^{m}y\in K$, by Theorem 2.5 it follows that $z_{1}xa^{m}y\leq z_{2}(z_{1}x)$ for some $z_{2}\in K$. Therefore

[TABLE]

Similarly, for $a^{m},yz_{2}z_{1}\in K$ there is $z_{3}\in K$ such that

[TABLE]

and for $a^{m},z_{1}xz_{3}\in K$ there is $z_{4}\in K$ such that

[TABLE]

Thus from (3) we obtain that

[TABLE]

Hence $xa^{m}y\in(xa^{m}ySya^{m}x]$ and so $xa^{m}y\in(xa^{m}ySya^{m}x]\cap(xa^{m}ySxa^{m}y]$, from (12).

To show the second condition choose $a\in S$ and $b\in Reg_{\leq}(S)$ be such that $a\leq b$. By the regularity of $b$ yields that $b\leq b(tb)^{n}$ for some $t\in S$ and for all $n\in\mathbb{N}$. Then there is $r\in\mathbb{N}$ such that $(tb)^{r}\in K$. Since $K$ ia an ideal $b(tb)^{r}\in K$ and so $b\in K$. Thus $a\in K$. Since $K$ is left Clifford ordered semigroup, $\mathcal{L}$ is congruence on $S$, by Theorem 2.5(i). Thus $a\mathcal{L}ab$. Then $a,ab$ are in Theorem 3.1, $a\leq azab$ for some $z\in S$. This proves the necessary condition.

Conversely, suppose that given conditions hold in $S$. Let $a\in S$. Then by the first condition there exists $n\in\mathbb{N}$ such that $a^{n+2}\leq a^{n+2}xa^{n+2},\;\textrm{for some}\;x\in S$. Thus $Reg_{\leq}(S)\neq\phi$. Say $T=Reg_{\leq}(S)$. Now for each $a\in S,\;\textrm{there exists}\;m\in\mathbb{N}$ such that $a^{m}\in T$. Let $s\in S$ and $x\in T$. Then for all $n\in\mathbb{N}\;\textrm{and for some}\;t\in S,\;x\leq(xt)^{n}x$. This implies $sx\leq sx(tx)^{n-1}tx,\;\textrm{for all}\;n\in\mathbb{N}$.

Then by the first condition there are $s_{1}\in S\;\textrm{and}\;m_{1}\in\mathbb{N}$ such that

[TABLE]

Therefore $sx\in T$.

We now show that $T$ is a left Clifford ordered semigroup. For this let us assume that $a,b\in T$. Then there is $t_{1}\in S$ such that

[TABLE]

Then by first condition there are $t_{2}\in S\;\textrm{and}\;m^{\prime}\in\mathbb{N}$ such that

[TABLE]

Therefore $ab\leq t^{\prime}_{1}a;\;\textrm{where}\;t^{\prime}_{1}=a(bt_{1}a)^{m^{\prime}}bt_{1}abt_{2}bt_{1}ab(bt_{1}a)^{m^{\prime}}.$ Since $a\in T$ we have $t_{1}^{{}^{\prime}}\in T$. Hence $T$ is left Clifford ordered semigroup.

Next let $a\in S\;\textrm{and}\;b\in T$ such that $a\leq b$. Using second condition we have $a\leq azab$, for some $z\in S$. Then for all $n\in\mathbb{N,}\;a\leq a(za)^{n}b^{n}$. So for some $m^{\prime\prime}\in\mathbb{N},\;(za)^{m^{\prime\prime}}\in T$. Since $T$ is a left Clifford, $(za)^{m^{\prime\prime}}b^{m^{\prime\prime}}\in(Ta]$. So $a\in(aSa]$, that is $a\in T$.

Finally to show $T$, an ideal of $S$ we need only to show that $xs\in T$. The regularity of $x$ yields that $xs\leq x(tx)^{n}s$ for all $n\in\mathbb{N}$. Also for some $l\in\mathbb{N}$, $x(tx)^{l}s\in T$. Then $xs\leq x(tx)^{l}s$ implies that $xs\in T$ by above. Thus $T$ is an ideal of $S$. Hence $S$ is a nil extension of a left Clifford ordered semigroup $T$. ∎