§ 2. Identity element; cancellable elements; invertible elements

1. Identity element

There exists at most one identity element under a given law $\top$, for if $e$ and $e'$ are identity elements then $e = e \top e' = e'$. An identity element is permutable with every element: it is a central element.

If $E, E'$ are monoids, a unital morphism of $E$ into $E'$ is called a monoid homomorphism or a monoid morphism of $E$ into $E'$.

If $(x_\alpha)_{\alpha \in \varnothing}$ is the empty family of elements of E its composition $e$ is also denoted by $\prod_{\alpha \in \varnothing} x_\alpha$. For example, we write

Let E be a unital magma whose law is denoted by $\top$ and $e$ its identity element. The support of a family $(x_i)_{i \in I}$ of elements of E is the set of indices $i \in I$ such that $x_i \neq e$. Let $(x_i)_{i \in I}$ be a family of elements of E with finite support. We shall define the composition $\prod_{i \in I} x_i$ in the two following cases:

(a) the set I is totally ordered; (b) E is associative and the $x_i$ are pairwise permutable.

In these two cases let S be the support of the family $(x_i)$. If J is a finite subset of I containing S, then $\prod_{i \in J} x_i = \prod_{i \in S} x_i$, as is seen by induction on the number of elements in J, applying Theorem 1 of § 1 in case (a) and Theorem 3 of § 1 in case (b). Let $\prod_{i \in I} x_i$ denote the common value of the compositions $\prod_{i \in J} x_i$ for all finite subsets of I containing S. When I is the interval $[p, \to]$ of $\mathbf{N}$, we also write $\prod_{i = p}^\infty x_i$.

With these definitions and notation, Theorems 1 and 3 of § 1 and the remarks following Theorem 3 extend to families with finite support.

The identity element of a law written additively is usually denoted by 0 and called zero or the null element (or sometimes the origin). Under a law written multiplicatively it is usually denoted by 1 and called the unit element (or unit).

2. Cancelable elements

On passing to the opposite law, left translations become right translations and conversely.

Let $\gamma_a, \delta_a$ (or $\gamma(a), \delta(a)$) denote the left and right translations by $a \in E$; then

In other words, for $a$ to be cancellable under the law $\mathsf{T}$, it is necessary and sufficient that each of the relations $a \mathsf{T} x = a \mathsf{T} y, x \mathsf{T} a = y \mathsf{T} a$ imply $x = y$ (it is said that "a can be cancelled" from each of these equalities). If there exists an identity element $e$ under the law $\mathsf{T}$, it is cancellable under this law: the translations $\gamma_e$ and $\delta_e$ are then the identity mapping of $E$ onto itself.

3. Invertible elements

An element $x$ of $E$ is called left invertible (resp. right invertible, resp. invertible) if it has a left inverse (resp. right inverse, resp. inverse).

A monoid all of whose elements are invertible is called a group.

Symmetric and symmetrizable are sometimes used instead of inverse and invertible. When the law on E is written additively, we generally say negative instead of inverse.

4. Monoid of fractions of a commutative monoid

In this no., e will denote the identity element of a monoid and x* the inverse of an invertible element x of E.

Let E be a commutative monoid, S a subset of E and S' the submonoid of E generated by S.

For $a \in E$ and $p \in S'$ the class of $(a, p)$ modulo $R$ is in general denoted by $a/p$ and called the fraction with numerator $a$ and denominator $p$. Then by definition $(a/p).(a'/|p') = aa'/|pp'$. The fractions $a/p$ and $a'/|p'$ are equal if and only if there exists $s$ in $S'$ with $spa' = sp'a$; if so, there exist $\sigma$ and $\sigma'$ in $S'$ with $a\sigma = a'\sigma'$ and $p\sigma = p'\sigma'$. In particular, $a/p = sa/sp$ for $a \in E$ and $s, p$ in $S'$. The identity element of $E_S$ is the fraction $e/e$.

Let $a/e = \varepsilon(a)$ for all $a \in E$. The above shows that $\varepsilon$ is a homomorphism of $E$ into $E_S$, called canonical. For all $p \in S'$, $(p/e).(e/p) = e/e$ and hence $e/p$ is the inverse of $\varepsilon(p) = p/e$; every element of $\varepsilon(S')$ is therefore invertible. Then $a/p = (a/e).(e/p)$, whence

5. Applications: I. Rational integers

Consider the commutative monoid $\mathbf{N}$ of natural numbers with law of composition addition; all the elements of $\mathbf{N}$ are cancellable under this law (Set Theory, III, § 5, no. 2, Corollary 3). The group of differences of $\mathbf{N}$ is denoted by $\mathbf{Z}$; its elements are called the rational integers; its law is called addition of rational integers and also denoted by +. The canonical homomorphism from $\mathbf{N}$ to $\mathbf{Z}$ is injective and we shall identify each element of $\mathbf{N}$ with its image in $\mathbf{Z}$. The elements of $\mathbf{Z}$ are by definition the equivalence classes determined in $\mathbf{N} \times \mathbf{N}$ by the relation between $(m_1, n_1)$ and $(m_2, n_2)$ which is written $m_1 + n_2 = m_2 + n_1$; an element $m$ of $\mathbf{N}$ is identified with the class consisting of the elements $(m + n, n)$, where $n \in \mathbf{N}$; it admits as negative in $\mathbf{Z}$ the class of elements $(n, m + n)$. Every element $(p, q)$ of $\mathbf{N} \times \mathbf{N}$ may be written in the form $(m + n, n)$ if $p \geq q$ or in the form $(n, m + n)$ if $p \leq q$; it follows that $\mathbf{Z}$ is the union of $\mathbf{N}$ and the set of negatives of the elements of $\mathbf{N}$. The identity element 0 is the only element of $\mathbf{N}$ whose negative belongs to $\mathbf{N}$.

For every natural number $m$, $-m$ denotes the negative rational integer of $m$ and $-\mathbf{N}$ denotes the set of elements $-m$ for $m \in \mathbf{N}$. Then

for $m \in \mathbf{N}$, $m = -m$ if and only if $m = 0$.

Let $m$ and $n$ be two natural numbers;

(a) if $m \geq n$, then $m + (-n) = p$, where $p$ is the element of $\mathbf{N}$ such that $m = n + p$; (b) if $m \leq n$, then $m + (-n) = -p$, where $p$ is the element of $\mathbf{N}$ such that $m + p = n$; (c) $(-m) + (-n) = -(m + n)$.

Properties (b) and (c) follow from no. 3, Proposition 4; as $\mathbf{Z} = \mathbf{N} \cup (-\mathbf{N})$, addition in $\mathbf{N}$ and properties (a), (b) and (c) describe completely addition in $\mathbf{Z}$.

More generally $-x$ is used to denote the negative of an arbitrary rational integer $x$; the composition $x + (-y)$ is abbreviated to $x - y$ (cf. no. 8).

The order relation $\leq$ between natural numbers is characterized by the following property: $m \leq n$ if and only if there exists an integer $p \in \mathbf{N}$ such that $m + p = n$ (Set Theory, III, § 3, no. 6, Proposition 13 and § 5, no. 2, Proposition 2). The relation $y - x \in \mathbf{N}$ between rational integers $x$ and $y$ is a total order relation on $\mathbf{Z}$ which extends the order relation $\leq$ on $\mathbf{N}$. For, for all $x \in \mathbf{Z}$, $x - x = 0 \in \mathbf{N}$; if $y - x \in \mathbf{N}$ and $z - y \in \mathbf{N}$, then

for $\mathbf{N}$ is stable under addition; if $y - x \in \mathbf{N}$ and $x - y \in \mathbf{N}$, then $y - x = 0$, for 0 is the only element of $\mathbf{N}$ whose negative belongs to $\mathbf{N}$; for arbitrary rational integers $x$ and $y$, $y - x \in \mathbf{N}$ or $x - y \in \mathbf{N}$, for $\mathbf{Z} = \mathbf{N} \cup (-\mathbf{N})$; finally, if $x$ and $y$ are natural numbers, then $y - x \in \mathbf{N}$ if and only if there exists $p \in \mathbf{N}$ such that $x + p = y$. This order relation is also denoted by $\leq$.

Henceforth when $\mathbf{Z}$ is considered as an ordered set, it will always be, unless otherwise mentioned, with the ordering that has just been defined, the natural numbers are identified with the integers $\geq 0$; they are also called positive integers; the integers $\leq 0$, negatives of the positive integers, are called negative integers; the integers $> 0$ (resp. $< 0$) are called strictly positive (resp. strictly negative); the set of integers $> 0$ is sometimes denoted by $\mathbf{N}^*$.

Let $x$, $y$ and $z$ be three rational integers; then $x \leq y$ if and only if x + z \leq y + z. For x - y = (x + z) - (y + z). This property is expressed by saying that the order relation on $\mathbf{Z}$ is invariant under translation.

6. Applications: II. Multiplication of rational integers