Chapter 1
Definitions and Notations

Let $X$ be a nonempty set and $\rho$, a quasi-order (a reflexive and transitive relation) on $X$. The pair $\left(X,\rho \right)$ is called a quasi-ordered set. For every $x\in X$, the set

is described as the (principal) $\rho$-ideal of $\left(X,\rho \right)$ generated by $x$. The relation $\rho \cap {\rho }^{-1}$ which is clearly an equivalence relation on $X$, is termed as the equivalence relation associated with $\rho$. An element $x\in X$ is said to be universally maximal in $X$ if

Let $\left(X,\rho \right)$ and $\left(X\prime ,\rho \prime \right)$ be quasi-ordered sets. Then,

is a quasi-order on $X\times X\prime$. The quasi-ordered set $\left(X\times X\prime ,\rho \times \rho \prime \right)$ is called product ordered set and is denoted by

Let $f:\left(X,\rho \right)\to \left(X\prime ,\rho \prime \right)$ be a mapping. The image of $x$ under $f$ is denoted by $xf$ (or by $\left(x\right)f$). $f$ is called an order preserving map or an order morphism if

for all $x,y\in X$.

If $f:\left(X,\rho \right)\to \left(X\prime ,\rho \prime \right)$ and $g:\left(X\prime ,\rho \prime \right)\to \left(X\prime \prime ,\rho \prime \prime \right)$ then $f\circ g$ denotes the composition

An order morphism $f:\left(X,\rho \right)\to \left(X\prime ,\rho \prime \right)$ is called a retraction (co-retraction) if there exists an order morphism $g:\left(X\prime ,\rho \prime \right)\to \left(X,\rho \right)$ such that

where $I_X$ denotes the identity map on $X$. (We shall use this notation for identity maps throughout this work).

An order morphism which is both a retraction and a co-retraction is called an order isomorphism (or isomorphism when there is no confusion). An order isomorphism of $\left(X,\rho \right)$ onto $\left(X\prime ,\rho \prime \right)$ may also be defined as a bijection satisfying the following condition: For all $x,y\in X$,

We regard composition of functions as a particular case of composition of relations defined on p. 13 of [2], so that $f\circ g$ is defined even when range $f\ne domg$ (so that $f\circ g=\varnothing$ if range $f\cap domg=\varnothing$), being written in the order shown in (1.5) which agrees with the definition given in [2].

All notations associated with semigroups, not explicitly defined here, are from [2] and [3]. We give below some additional notations which are used throughout this work.

By a semigroup we mean a regular semigroup which we denote by $S$. For $x,y\in S$ we denote by $R_x$ and $L_y$ the $\mathcal{ℛ}$ and $\mathcal{ℒ}$-classes of $x$ and $y$ respectively. We write

so that if $H_{x,y}\ne \varnothing$, it is an $\mathcal{\mathscr{H}}$-class of $S$. Further, if $X\subseteq S$, we define

so that $E\left(S\right)$ denotes the set of idempotents of $S$, and when there is no confusion, we write $E$ instead of $E\left(S\right)$. On $E$, define

Clearly ${\omega }^r\left({\omega }^l\right)$ denotes the quasi-order on $E$ induced by the inclusion relation among the principal right (left) ideals. Thus for all $e\in E$,

and

It can be easily verified that

and in particular, $\omega$ is a partial order.

On $S$, define

and for $x\in S$ and $X\subseteq S$,

For convenience, denote $i\left(x;S\right)$ by $i\left(x\right)$. If $e,f\in E$ and $x\in H_{e,f}$, then $i\left(x;f,e\right)$ is the unique inverse of $x$ in $H_{f,e}$.

Let $L$ and $L\prime \left(R\text{ and }R\prime \right)$ be two $\mathcal{ℒ}\left(\mathcal{ℛ}\right)$-classes of $S$. If the right (left) translation ${\rho }_x\left({\lambda }_x\right)$ determined by $x\in S$ is a one-to-one $\mathcal{ℛ}\left(\mathcal{ℒ}\right)$-class preserving map of $L$ onto $L\prime \left(R\text{ onto }R\prime \right)$, then we say that $x$ translates $L$ onto $L\prime \left(R\text{ onto }R\prime \right)$ so that $L_x=L\prime \left(xR=R\prime \right)$. Clearly, if $x$ translates $L$ onto $L\prime$ and if $y$ translates $L\prime$ onto $L\prime \prime$, then $xy$ translates $L$ onto $L\prime \prime$ and by the associativity of multiplication in $S$,

In this investigation we deal with the left–right duals of statements and concepts. Left–right dual of a statement or concept $T$ is denoted by $T^{\ast }$ while the actual statement or definition of $T^{\ast }$ is omitted. Thus the quasi-order ${\omega }^l$ is the left–right dual of ${\omega }^r$. Corresponding to any statement or concept involving ${\omega }^r$, there is a statement or concept involving ${\omega }^l$ which will be the left–right dual of the former. The same remark applies to definitions of biordered sets (Definition 2.1), bimorphisms (Definition 2.2) etc., where the quasi-order ${\omega }^r$, function ${\tau }^r\left(e\right)$ etc., are regarded as duals of ${\omega }^l,{\tau }^l\left(e\right)$ etc. Unless otherwise stated, with every statement, axiom or definition we assume its dual and its explicit formulation is omitted.