§ 1. Structures and isomorphisms

1. Structures and isomorphisms

The purpose of this chapter is to describe once and for all a certain number of formative constructions and proofs (cf. Chapter I, § 1, no. 3 and § 2, no. 2) which arise very frequently in mathematics.

1. Echelons

An echelon construction scheme is a sequence $c_1$ , $c_2$ , ..., $c_m$ of ordered pairs of natural integers [^1] $c_i = (a_i,\ b_i)$ , satisfying the following conditions :

(a)　If $b_i = 0$ , then $1 \leqslant a_i \leqslant i - 1$ . (b)　If $a_i \neq 0$ and $b_i \neq 0$ , then $1 \leqslant a_i \leqslant i - 1$ and $1 \leqslant b_i \leqslant i - 1$ .

These conditions imply that $c_1 = (0,\ b_1)$ , with $b_1 > 0$ . If $n$ is the largest of the integers $b_i$ which appear in the pairs $(0,\ b_i)$ , then $c_1, c_2, ..., c_m$ is said to be an echelon construction scheme on $n$ terms.

Given an echelon construction scheme $\mathrm{S} = (c_1,\ c_2,\ \ldots,\ c_m)$ on $n$ terms, and given $n$ terms $\mathrm{E}_1$ , $\mathrm{E}_2$ , ..., $\mathrm{E}_n$ in a theory $\mathscr{T}$ which is stronger than the theory of sets, an echelon construction of scheme $\mathrm{S}$ on $\mathrm{E}_1$ , ..., $\mathrm{E}_n$ is defined to be a sequence $\mathrm{A}_1$ , $\mathrm{A}_2$ , ..., $\mathrm{A}_m$ of $m$ terms in the theory $\mathscr{T}$ , defined step by step by the following conditions :

(a)　If $c_i = (0,\ b_i)$ , then $\mathrm{A}_i$ is the term $\mathrm{E}_{b_i}$ . (b)　If $c_i = (a_i,\ 0)$ , then $\mathrm{A}_i$ is the term $\mathfrak{P}(\mathrm{A}_{a_i})$ . (c)　If $c_i = (a_i,\ b_i)$ , where $a_i \neq 0$ and $b_i \neq 0$ , then $\mathrm{A}_i$ is the term $\mathrm{A}_{a_i} \times \mathrm{A}_{b_i}$ .

The last term $\mathrm{A}_m$ of the echelon construction of scheme S on $\mathrm{E}_1$ , ..., $\mathrm{E}_n$ is called the echelon of scheme S on the base sets $\mathrm{E}_1$ , ..., $\mathrm{E}_n$ ; in the general arguments which follow, it will be denoted by the notation $\mathrm{S}(\mathrm{E}_1, \ldots, \mathrm{E}_n)$ .

Example 03V4

Given two sets E, F, the set $\mathfrak{P}(\mathfrak{P}(\mathrm{E})) \times \mathfrak{P}(\mathrm{F})$ is an echelon on E, F, with scheme

(0, 1), \quad (0, 2), \quad (1, 0), \quad (3, 0), \quad (2, 0), \quad (4, 5).

It is also the echelon on E, F with scheme

(0, 2), \quad (0, 1), \quad (1, 0), \quad (2, 0), \quad (4, 0), \quad (5, 3).

Distinct schemes may therefore give rise to the same echelon on the same terms.

2. Canonical extensions of mappings

Let $\mathrm{S} = (c_1, c_2, \ldots, c_m)$ be an echelon construction scheme on $n$ terms. Let $\mathrm{E}_1$ , ..., $\mathrm{E}_n$ , $\mathrm{E}'_1$ , ..., $\mathrm{E}'_n$ be sets (terms in $\mathscr{T}$ ) and let $f_1$ , ..., $f_n$ be terms in $\mathscr{T}$ such that the relations " $f_i$ is a mapping of $\mathrm{E}_i$ into $\mathrm{E}'_i$ " are theorems in $\mathscr{T}$ for $1 \leqslant i \leqslant n$ . Let $\mathrm{A}_1$ , ..., $\mathrm{A}_m$ (resp. $\mathrm{A}'_1$ , ..., $\mathrm{A}'_m$ ) be the echelon construction of scheme S on $\mathrm{E}_1$ , ..., $\mathrm{E}_n$ (resp. $\mathrm{E}'_1$ , ..., $\mathrm{E}'_n$ ). We define step by step a sequence of $m$ terms $g_1$ , ..., $g_m$ such that $g_i$ is a mapping of $\mathrm{A}_i$ into $\mathrm{A}'_i$ (for $1 \leqslant i \leqslant m$ ) by the following conditions :

(a) If $c_i = (0, b_i)$ , so that $\mathrm{A}_i = \mathrm{E}_{b_i}$ and $\mathrm{A}'_i = \mathrm{E}'_{b_i}$ , then $g_i$ is the mapping $f_{b_i}$ .

(b) If $c_i = (a_i, 0)$ , so that $\mathrm{A}_i = \mathfrak{P}(\mathrm{A}_{a_i})$ and $\mathrm{A}'_i = \mathfrak{P}(\mathrm{A}'_{a_i})$ , then $g_i$ is the canonical extension $\hat{g}_{a_i}$ of $g_{a_i}$ to sets of subsets (Chapter II, § 5, no. 1).

\mathrm{A}_i = \mathrm{A}_{a_i} \times \mathrm{A}_{b_i} \qquad \text{and} \qquad \mathrm{A}'_i = \mathrm{A}'_{a_i} \times \mathrm{A}'_{b_i},

then $g_i$ is the canonical extension $g_{a_i} \times g_{b_i}$ of $g_{a_i}$ and $g_{b_i}$ to $\mathrm{A}_{a_i} \times \mathrm{A}_b$ (Chapter II, § 3, no. 9).

The last term $g_m$ of this sequence is called the canonical extension, with scheme S, of the mappings $f_1$ , ..., $f_n$ , and will be denoted by $\langle f_1, \ldots, f_n \rangle^{\mathrm{S}}$ .

¶ The following criteria can be verified step by step :

CST1. If $f_i$ is a mapping of $\mathrm{E}_i$ into $\mathrm{E}'_i$ , and if $f'_i$ is a mapping of $\mathrm{E}'_i$ into $\mathrm{E}''_i$ ( $1 \leqslant i \leqslant n$ *), then for every echelon construction scheme* S on $n$ terms we have

\langle f'_1 \circ f_1, f'_2 \circ f_2, \ldots, f'_n \circ f_n \rangle^{\mathrm{S}} = \langle f'_1, f'_2, \ldots, f'_n \rangle^{\mathrm{S}} \circ \langle f_1, f_2, \ldots, f_n \rangle^{\mathrm{S}}.

CST2. If $f_i$ is injective (resp. surjective) for $1 \leqslant i \leqslant n$ , then $\langle f_1, \ldots, f_n \rangle^{\mathrm{S}}$ is injective (resp. surjective).

This criterion follows from the corresponding properties of the extension $\hat{g}$ (Chapter II, § 5, no. 1, Proposition 1) and the extension $g \times h$ (Chapter II, § 3, no. 9).

CST3. If $f_i$ is a bijection of $\mathrm{E}_i$ onto $\mathrm{E}'_i$ , and if $f_i^{-1}$ is the inverse bijection [^2], then $\langle f_1, \ldots, f_n \rangle^{\mathrm{S}}$ is a bijection and $\langle f_1^{-1}, \ldots, f_n^{-1} \rangle^{\mathrm{S}}$ is its inverse; in other words,

(\langle f_1, \ldots, f_n \rangle^{\mathrm{S}})^{-1} = \langle f_1^{-1}, \ldots, f_n^{-1} \rangle^{\mathrm{S}}.

This follows immediately from CST1 and CST2.

3. Transportable relations

Let $\mathscr{T}$ be a theory which is stronger than the theory of sets, let $x_1, \ldots, x_n$ , $s_1, \ldots, s_p$ be distinct letters which are distinct from the constants of $\mathscr{T}$ , and let $\mathrm{A}_1, \ldots, \mathrm{A}_m$ be terms in $\mathscr{T}$ in which none of the letters $x_i$ $(1 \leqslant i \leqslant n)$ and $s_j$ $(1 \leqslant j \leqslant p)$ appears. Let $\mathrm{S}_1, \ldots, \mathrm{S}_p$ be echelon construction schemes on $n + m$ terms. Then the relation $\mathrm{T}\{x_1, \ldots, x_n, s_1, \ldots, s_p\}$ :

" $s_1 \in \mathrm{S}_1(x_1, \ldots, x_n, \mathrm{A}_1, \ldots, \mathrm{A}_m)$ and $s_2 \in \mathrm{S}_2(x_1, \ldots, x_n, \mathrm{A}_1, \ldots, \mathrm{A}_m)$ and $\ldots$ and $s_p \in \mathrm{S}_p(x_1, \ldots, x_n, \mathrm{A}_1, \ldots, \mathrm{A}_m)$ "

is called a typification of the letters $s_1, \ldots, s_p$ .

¶ Let $\mathrm{R}\{x_1, \ldots, x_n, s_1, \ldots, s_p\}$ be a relation in $\mathscr{T}$ which contains certain of the letters $x_i$ , $s_j$ (and possibly other letters as well). Then R is said to be transportable (in $\mathscr{T}$ ) with respect to the typification T, the $x_i$ $(1 \leqslant i \leqslant n)$ being considered as principal base sets and the $\mathrm{A}_h$ $(1 \leqslant h \leqslant m)$ as auxiliary base sets, if the following condition is satisfied : let $y_1, \ldots, y_n, f_1, \ldots, f_n$ be distinct letters which are distinct from the $x_i$ $(1 \leqslant i \leqslant n)$ , the $s_j$ $(1 \leqslant j \leqslant p)$ , the constants of $\mathscr{T}$ , and all the letters which appear in R or in the terms $\mathrm{A}_h$ $(1 \leqslant h \leqslant m)$ , and let $\mathrm{Id}_h$ $(1 \leqslant h \leqslant m)$ denote the identity mapping of $\mathrm{A}_h$ on to itself. Then the relation

(1) " $\mathrm{T}\{x_1, \ldots, x_n, s_1, \ldots, s_p\}$ and ( $f_1$ is a bijection of $x_1$ onto $y_1$ ) and $\ldots$ and ( $f_n$ is a bijection of $x_n$ onto $y_n$ )"

implies, in $\mathscr{T}$ , the relation

(2) $\mathrm{R}\{x_1, \ldots, x_n, s_1, \ldots, s_p\} \Leftrightarrow \mathrm{R}\{y_1, \ldots, y_n, s'_1, \ldots, s'_p\}$ , where

(3) \qquad s'_j = \langle f_1, \ldots, f_n, \mathrm{Id}_1, \ldots, \mathrm{Id}_m \rangle^s{}_j(s_j) \qquad (1 \leqslant j \leqslant p).

There is an analogous but simpler definition in the case where there is no auxiliary set.

For example, if $n = p = 2$ and if the typification T is " $s_1 \in x_1$ and $s_2 \in x_1$ ", the relation $s_1 = s_2$ is transportable. On the other hand, the relation $x_1 = x_2$ is not transportable.

4. Species of structures

Let $\mathscr{T}$ be a theory which is stronger than the theory of sets. A species of structures in $\mathscr{T}$ is a text $\Sigma$ formed of the following assemblies :

(1) a certain number of letters $x_1, \ldots, x_n, s$ , distinct from each other and from the constants of $\mathscr{T}$ ; $x_1, \ldots, x_n$ are called the principal base sets of the species of structures $\Sigma$ ;

(2) a certain number of terms $\mathrm{A}_1, \ldots, \mathrm{A}_m$ in $\mathscr{T}$ in which none of the letters $x_1, \ldots, x_n, s$ appears, and which are called the auxiliary base sets of $\Sigma$ ; $\Sigma$ possibly contains no auxiliary base sets (but it must contain at least one principal base set);

(3) a typification $\mathrm{T}\{x_1, \ldots, x_n, s\}$ :

s \in \mathrm{S}(x_1, \ldots, x_n, \mathrm{A}_1, \ldots, \mathrm{A}_m),

where S is an echelon construction scheme on $n + m$ terms (no. 1); $\mathrm{T}\{x_1, \ldots, x_n, s\}$ is called the typical characterization of the species of structures $\Sigma$ ;

(4) a relation $\mathrm{R}\{x_1, \ldots, x_n, s\}$ which is transportable (in $\mathscr{T}$ ) with respect to the typification T, the $x_i$ being the principal base sets and the $\mathrm{A}_h$ the auxiliary base sets (no. 3); R is called the axiom of the species of structures $\Sigma$ .

The theory $\mathscr{T}_\Sigma$ which has the same axiom schemes as $\mathscr{T}$ and whose explicit axioms are those of $\mathscr{T}$ , together with the axiom "T and R", is called the theory of the species of structures $\Sigma$ . The constants of $\mathscr{T}_\Sigma$ are therefore the constants of $\mathscr{T}$ and the letters which appear in T or in R.

¶ Let $\mathscr{T}'$ be a theory which is stronger than $\mathscr{T}$ , and let $\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{U}$ be terms in $\mathscr{T}'$ . In the theory $\mathscr{T}'$ , U is said to be a structure of species $\Sigma$ on the principal base sets $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , with $\mathrm{A}_1, \ldots, \mathrm{A}_m$ as auxiliary base sets, if the relation

\text{"}\mathrm{T}\{\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{U}\} \text{ and } \mathrm{R}\{\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{U}\}\text{"}

is a theorem in $\mathscr{T}'$ . When this is so, then for each theorem $\mathrm{B}\{x_1, \ldots, x_n, s\}$ in the theory $\mathscr{T}_\Sigma$ the relation $\mathrm{B}\{\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{U}\}$ is a theorem in $\mathscr{T}'$ (Chapter I, § 2, no. 3). In $\mathscr{T}_\Sigma$ , the constant $s$ is called the generic structure of the species $\Sigma$ .

¶ In the theory $\mathscr{T}'$ , the principal base sets $\mathrm{E}_1, \ldots, \mathrm{E}_n$ are said to be endowed with the structure $\mathrm{U}$ . Clearly, $\mathrm{U}$ is an element of the set

\mathrm{S}(\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{A}_1, \ldots, \mathrm{A}_m).

The set of elements $\mathrm{V}$ of $\mathrm{S}(\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{A}_1, \ldots, \mathrm{A}_m)$ which satisfy the relation $\mathrm{R}\{\mathrm{E}_1, \ldots, \mathrm{E}_n, \mathrm{V}\}$ is therefore the set of structures of the species $\Sigma$ on $\mathrm{E}_1, \ldots, \mathrm{E}_n$ (and it may be empty).

Examples

Example 1 03V5

s \circ s = s \quad \text{and} \quad s \cap \overset{-1}{s} = \Delta_{\mathbf{A}}

( $\Delta_{\mathbf{A}}$ being the diagonal of $\mathrm{A} \times \mathrm{A}$ ), which is a transportable relation with respect to the typification $s \in \mathfrak{P}(\mathrm{A} \times \mathrm{A})$ , as is easily verified. It is clear that the theory of this species of structures is just the theory of ordered sets (Chapter III, §1, no. 3); and therefore the species of structure so defined is also called the species of order structures on $\mathrm{A}$ . In Chapter III we saw many examples of sets endowed with structures of this species.

Example 2 03V6

Take $\mathscr{T}$ to be the theory of sets, and consider the species of structures which has no auxiliary base set, one principal base set $\mathrm{A}$ , the typical characterization $\mathrm{F} \in \mathfrak{P}((\mathrm{A} \times \mathrm{A}) \times \mathrm{A})$ , and as axiom the transportable relation "F is a functional graph whose domain is $\mathrm{A} \times \mathrm{A}$ ". The structures of this species are particular cases of what are called algebraic structures, and the function whose graph is $\mathrm{F}$ (a mapping of $\mathrm{A} \times \mathrm{A}$ into $\mathrm{A}$ ) is called the (everywhere defined) internal law of composition of such a structure.

Example 3 03V7

As before, let $\mathscr{T}$ be the theory of sets, and consider the species of structures which has no auxiliary base set, one principal base set $\mathrm{A}$ , the typical characterization $\mathrm{V} \in \mathfrak{P}(\mathfrak{P}(\mathrm{A}))$ , and as axiom the transportable relation

(\forall \mathrm{V}) \, ((\mathrm{V}' \subset \mathrm{V}) \Rightarrow ((\bigcup_{\mathrm{X} \in \mathrm{V}'} \mathrm{X}) \in \mathrm{V}))

\text{and} \quad (\forall \mathrm{X})(\forall \mathrm{Y})((\mathrm{X} \in \mathrm{V} \text{ and } \mathrm{Y} \in \mathrm{V}) \Rightarrow ((\mathrm{X} \cap \mathrm{Y}) \in \mathrm{V})).

This species of structures is called the species of topological structures. A structure of this species is also called a topology, and the relation $\mathrm{X} \in \mathrm{V}$ is expressed by saying that $\mathrm{X}$ is an open set in the topology $\mathrm{V}$ (General Topology, Chapter I, § 1).

Example 4 03V8

\* Take $\mathscr{T}$ to be the theory of the species of division ring structures, which has (among other things) a constant K as unique (principal) base set. The species of structure of a left vector space over K has K as auxiliary base set, a principal base set E, and as typical characterization the relation

V \in \mathfrak{P}((E \times E) \times E) \times \mathfrak{P}((K \times E) \times E)

(pr $_1$ V being the graph of addition in E, and pr $_2$ V the graph of scalar multiplication); we shall not state here the axiom for this species of structures.

Example 5 03V9

Again let $\mathscr{T}$ be the theory of sets; in this theory, the field $\mathbf{C}$ of complex numbers is a term which contains no letters. The species of structure of a complex analytic manifold of dimension n has $\mathbf{C}$ as auxiliary base set, and one principal base set V. We shall not indicate here the typical characterization or the axiom of this species of structure. \

Remarks

Remark 1 03VA

In applications it is often the case (as in Example 4 above) that the echelon $S(E_1, \ldots, E_n, A_1, \ldots, A_m)$ is a product of echelons

S_1(E_1, \ldots, A_m) \times \cdots \times S_p(E_1, \ldots, A_m).

If so, the letter $s$ in the definition of $\Sigma$ is often replaced by a " $p$ -tuple" $(s_1, \ldots, s_p)$ (cf. Chapter II, § 2, no. 1).

Moreover, the axiom of a species of structures $\Sigma$ is most frequently written as a conjunction of several transportable relations (as in Example 3 above). These relations are called the axioms of the species $\Sigma$ .

Remark 2 03VB

Names are given to the species of structures most frequently used in mathematics, and to sets endowed with structures of these species. Thus an ordered set (Chapter III, § 1) is a set endowed with an order structure (Example 1); \* in the later Books of this series, we shall define the notions of group, field, topological space, differentiable manifold, etc., all of which denote sets endowed with certain structures. \

Remark 3 03VC

By abuse of language, in the theory of sets $\mathscr{T}$ , the giving of $n$ distinct letters $x_1, \ldots, x_n$ (with no typical characterization and no axiom) is considered as a species of structure $\Sigma_0$ , called the structure of a set on the $n$ principal base sets $x_1, \ldots, x_n$ .

5. Isomorphisms and transport of structures

Let $\Sigma$ be a species of structures in a theory $\mathscr{T}$ , on $n$ principal base sets $x_1, \ldots, x_n$ , with $m$ auxiliary base sets $A_1, \ldots, A_m$ . Let S be the echelon construction scheme on $n + m$ letters which features in the typical characterization of $\Sigma$ , and let R be the axiom of $\Sigma$ . In a theory $\mathscr{T}'$ which is stronger than $\mathscr{T}$ , let U be a structure of species $\Sigma$ on sets $E_1, \ldots, E_n$ (as principal base sets) and let U $'$ be a structure of the same species on sets $E'_1, \ldots, E'_n$ . Finally, let $f_i$ (in $\mathscr{T}'$ ) be a bijection of $E_i$ onto $\mathrm{E}'_i$ $(1 \leqslant i \leqslant n)$ . Then $(f_1, \ldots, f_n)$ is said to be an isomorphism of the sets $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , endowed with the structure U, onto the sets $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ , endowed with the structure $\mathrm{U}'$ , if we have (in $\mathscr{T}'$ )

(4)

\langle f_1, \ldots, f_n, \mathrm{Id}_1, \ldots, \mathrm{Id}_m \rangle^{\mathrm{S}}(\mathrm{U}) = \mathrm{U}'

where $\mathrm{Id}_h$ denotes the identity mapping of $\mathrm{A}_h$ onto itself $(1 \leqslant h \leqslant m)$ .

¶ Let $f'_i$ be the inverse of the bijection $f_i$ $(1 \leqslant i \leqslant n)$ . It follows immediately from (4) and the criterion CST3 (no. 2) that we have

\langle f'_1, \ldots, f'_n, \mathrm{Id}_1, \ldots, \mathrm{Id}_m \rangle^{\mathrm{S}}(\mathrm{U}') = \mathrm{U}

and consequently that $(f'_1, \ldots, f'_n)$ is an isomorphism of $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ , endowed with $\mathrm{U}'$ , onto $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , endowed with U. The isomorphisms $(f_1, \ldots, f_n)$ and $(f'_1, \ldots, f'_n)$ are said to be inverses of each other.

$\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ , endowed with $\mathrm{U}'$ , are said to be isomorphic to $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , endowed with U, if there exists an isomorphism of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ onto $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ ; in this case the structures U and $\mathrm{U}'$ are said to be isomorphic.

¶ The above definitions, together with CST1, imply the following criterion:

CST4. Let U, $\mathrm{U}'$ , $\mathrm{U}''$ be three structures of the same species $\Sigma$ on the principal base sets $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ , $\mathrm{E}''_1, \ldots, \mathrm{E}''_n$ , respectively. Let $f_i$ be a bijection of $\mathrm{E}_i$ onto $\mathrm{E}'_i$ , and let $g_i$ be a bijection of $\mathrm{E}'_i$ onto $\mathrm{E}''_i$ $(1 \leqslant i \leqslant n)$ . If $(f_1, \ldots, f_n)$ and $(g_1, \ldots, g_n)$ are isomorphisms, then so is $(g_1 \circ f_1, \ldots, g_n \circ f_n)$ .

An isomorphism of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ onto $\mathrm{E}_1, \ldots, \mathrm{E}_n$ (with respect to the same structure) is called an automorphism of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ . The composition of two automorphisms of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ is an automorphism, and so is the inverse of an automorphism, \* so that the automorphisms of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ form a group. \*

Remark 03VD

By abuse of language, if $f_i$ is any bijection of $\mathrm{E}_i$ onto $\mathrm{E}'_i$ $(1 \leqslant i \leqslant n)$ , $(f_1, \ldots, f_n)$ is said to be an isomorphism of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ onto $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ with respect to the species of structure of a set (no. 4, Remark 3).

CST5. In a theory $\mathscr{T}'$ which is stronger than $\mathscr{T}$ , let U be a structure of species $\Sigma$ on $\mathrm{E}_1, \ldots, \mathrm{E}_n$ , and let $f_i$ be a bijection of $\mathrm{E}_i$ onto a set $\mathrm{E}'_i$ $(1 \leqslant i \leqslant n)$ . Then there exists a unique structure of species $\Sigma$ on $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ such that $(f_1, \ldots, f_n)$ is an isomorphism of $\mathrm{E}_1, \ldots, \mathrm{E}_n$ onto $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ .

For this structure, if it exists, can only be the term $\mathrm{U}'$ defined by the relation (4); it remains to be verified that this term is indeed a structure of species $\Sigma$ , i.e., that the relation $\mathrm{R}\{\mathrm{E}'_1, \ldots, \mathrm{E}'_n, \mathrm{U}'\}$ is true in $\mathscr{T}'$ . But this follows from the fact that $\mathrm{R}\{x_1, \ldots, x_n, s\}$ is transportable, for

$R\{E'_1, \ldots, E'_n, U'\}$ is equivalent in $\mathscr{T}'$ to the relation $R\{E_1, \ldots, E_n, U\}$ (no. 3), which is true in $\mathscr{T}'$ by hypothesis.

¶ The structure $U'$ is said to be obtained by transporting the structure $U$ to the sets $E'_1, \ldots, E'_n$ by means of the bijective mappings $f_1, \ldots, f_n$ . Thus two structures of the same species are isomorphic if and only if each is obtained from the other by transport of structure.

It may happen that any two structures of species $\Sigma$ are necessarily isomorphic; the species of structure $\Sigma$ is then said be univalent. \* This is the case for the structure of an infinite cyclic group (isomorphic to $\mathbf{Z}$ ), the structure of a prime field of characteristic zero (isomorphic to $\mathbf{Q}$ ), the structure of a complete Archimedean ordered field (isomorphic to $\mathbf{R}$ ), the structure of an algebraically closed, connected, locally compact topological field (isomorphic to $\mathbf{C}$ ), and of the structure of a non-commutative, connected, locally compact topological division ring (isomorphic to $\mathbf{H}$ , the division ring of quaternions). For some of these species of structure, for example that of a prime field of characteristic zero, or that of a complete Archimedean ordered field, there is not even any automorphism other than the identity mapping; but there do exist such automorphisms for the other examples given above (for example, the symmetry $x \rightarrow -x$ in $\mathbf{Z}$ ). \

It will be observed that the above species of structure are essentially those which are the basis of classical mathematics. On the other hand, \* the species of group structures, the species of structures of an ordered set, and the species of topological structures, are not univalent. \

6. Deduction of structures

Let $\Sigma$ be a species of structures in a theory $\mathscr{T}$ , on $n$ principal base sets $x_1, \ldots, x_n$ , with $m$ auxiliary base sets $A_1, \ldots, A_m$ . Let $s$ be the generic structure of $\Sigma$ , and let $T$ be an echelon construction scheme on $n + m$ terms. A term $V\{x_1, \ldots, x_n, s\}$ which contains no letter other than the constants of $\mathscr{T}_\Sigma$ is said to be intrinsic for $s$ , of type $T(x_1, \ldots, x_n, A_1, \ldots, A_m)$ , if it satisfies the following conditions :

(1) the relation $V\{x_1, \ldots, x_n, s\} \in T(x_1, \ldots, x_n, A_1, \ldots, A_m)$ is a theorem in $\mathscr{T}_\Sigma$ ;

(2) let $\mathscr{T}'_\Sigma$ be the theory obtained by adjoining to the axioms of $\mathscr{T}_\Sigma$ the axioms " $f_i$ is a bijection of $x_i$ onto $y_i$ " $(1 \leqslant i \leqslant n)$ (where the letters $y_i$ and $f_i$ are distinct from each other and from the constants of $\mathscr{T}_\Sigma$ , for $1 \leqslant i \leqslant n$ ); if $s'$ is the structure obtained by transporting $s$ by means of $(f_1, \ldots, f_n)$ (no. 5), then

V\{y_1, \ldots, y_n, s'\} = \langle f_1, \ldots, f_n. \mathrm{Id}_1, \ldots, \mathrm{Id}_m \rangle^{\mathrm{T}}(V\{x_1, \ldots, x_n, s\})

is a theorem in $\mathscr{T}'_\Sigma$ .

Most of the terms which one is led to define in the theory of a species of structures are intrinsic terms.

¶ Let $\Theta$ be another species of structures in the theory $\mathscr{T}$ , on $r$ principal base sets $u_1,\ldots,u_r$ , with $p$ auxiliary base sets $B_1,\ldots,B_p$ , and let $t\in T(u_1,\ldots,u_r,B_1,\ldots,B_p)$ be the typical characterization of $\Theta$ (no. 4). Then a system of $r+1$ terms $P,U_1,\ldots,U_r$ , intrinsic for $s$ , and such that $P$ is a structure of species $\Theta$ on $U_1,\ldots,U_r$ , in the theory $\mathscr{T}_\Sigma$ , is called a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$ . By abuse of language, the term $P$ alone is often called a procedure of deduction.

¶ Let $\mathscr{T}'$ be a theory stronger than $\mathscr{T}$ . If $\mathscr{S}$ is a structure in $\mathscr{T}'$ of species $\Sigma$ on $E_1,\ldots,E_n$ , then $P\{E_1,\ldots,E_n,\mathscr{S}\}$ is a structure of species $\Theta$ on the $r$ sets $F_j=U_j\{E_1,\ldots,E_n,\mathscr{S}\}$ $(1\leq j\leq r)$ , said to be deduced from $\mathscr{S}$ by the procedure $P$ , or subordinate to $\mathscr{S}$ . The hypothesis that the terms $P,U_1,\ldots,U_r$ are intrinsic for $s$ moreover implies the following criterion:

CST6. Let $(g_1,\ldots,g_n)$ be an isomorphism of $E_1,\ldots,E_n$ , endowed with a structure $\mathscr{S}$ of species $\Sigma$ , onto $E'_1,\ldots,E'_n$ , endowed with a structure $\mathscr{S}'$ of the same species. If $U_j$ is of type $\mathfrak{P}(T_j)$ , put

h_j=\langle g_1,\ldots,g_n,\mathrm{Id}_1,\ldots,\mathrm{Id}_{m}\rangle^{T_j}\quad(1\leq j\leq r),

and let $F'_j=U_j\{E'_1,\ldots,E'_n,\mathscr{S}'\}$ $(1\leq j\leq r)$ . Then $(h_1,\ldots,h_r)$ is an isomorphism of $F_1,\ldots,F_r$ onto $F'_1,\ldots,F'_r$ when these systems of sets are endowed with the structures of species $\Theta$ deduced from $\mathscr{S}$ and $\mathscr{S}'$ respectively by the procedure $P$ .

It is clear that the terms $x_1,\ldots,x_n$ are intrinsic for $s$ . In many cases, the terms $U_1,\ldots,U_r$ are certain of the letters $x_1,\ldots,x_n$ ; the structure of species $\Theta$ deduced from $s$ by the procedure $P$ is then said to be a structure underlying $s$ .

Examples

Example 1 03VE

\* The species of topological group structures has a single principal base set $A$ , no auxiliary base set, and the corresponding generic structure is a pair $(s_1,s_2)$ ( $s_1$ being the graph of the law of composition on $A$ , and $s_2$ the set of open sets in the topology of $A$ ; cf. General Topology, Chapter III, § 1). Each of the terms $s_1,s_2$ is a procedure of deduction and provides respectively the group structure and the topology underlying the topological group structure $(s_1,s_2)$ .

Likewise, from a vector space structure can be deduced an underlying commutative group structure. From a ring structure can be deduced an underlying commutative group structure and a (multiplicative) semigroup

IV STRUCTURES

structure. From the structure of a differentiable manifold can be deduced an underlying topology, etc.

Example 2 03VF

The species of vector space structures over $\mathbf{C}$ (resp. $\mathbf{R}$ ) has a principal base set $E$ , an auxiliary base set equal to $\mathbf{C}$ (resp. $\mathbf{R}$ ), and typical characterization

s_1\in\mathfrak{P}((E\times E)\times E)\qquad\text{and}\qquad s_2\in\mathfrak{P}((\mathbf{C}\times E)\times E)

(resp. $s_1\in\mathfrak{P}((E\times E)\times E)$ and $s_2\in\mathfrak{P}((\mathbf{R}\times E)\times E)$ ).

The pair $(s_1,s_2\cap((\mathbf{R}\times E)\times E))$ is a procedure of deduction of a vector space structure over $\mathbf{R}$ from a vector space structure over $\mathbf{C}$ (“restriction of the field of scalars to $\mathbf{R}$ ”). \

Example 3 03VG

Suppose that $\Theta$ has the same (principal and auxiliary) base sets as $\Sigma$ , and the same typical characterization. If, moreover, the axiom of $\Sigma$ implies (in $\mathscr{T}$ ) the axiom of $\Theta$ , it is clear that the term $s$ is a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$ . $\Theta$ is then said to be poorer than $\Sigma$ , and $\Sigma$ is richer than $\Theta$ . Every structure of species $\Sigma$ , in a theory $\mathscr{T}'$ which is stronger than $\mathscr{T}$ , is then also a structure of species $\Theta$ . For example, the species of structures of totally ordered sets (obtained by taking as axiom the conjunction of the axiom of order structures (no. 4, Example 1) and the relation $s\cup s^{-1}=A\times A$ ) is richer than the species of order structures. \*The species of commutative group structures is richer than the species of group structures. The species of compact topological space structures is richer than the species of topological structures, etc. \

Example 4

\* When each of $\Sigma$ and $\Theta$ is the species of group structures (resp. ring structures), there is defined in algebra a procedure of deduction which associates with each group structure (resp. ring structure) the group structure (resp. ring structure) on its centre. When $\Sigma$ is the species of vector space structures over a field $K$ , and when $\Theta$ is the species of algebraic structures over $K$ , there are defined procedures of deduction which associate with every vector space over $K$ its tensor algebra or its exterior algebra. We shall meet many other examples later in this series. \

Remark 03VH

When $P$ is a “q-tuple” $(P_1,\ldots,P_q)$ , it is also said that the terms $P_1,\ldots,P_q$ constitute a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$ .

7. Equivalent species of structures

Let $\Sigma$ and $\Theta$ be two species of structures, in the same theory $\mathscr{T}$ , having the same principal base sets $x_1,\ldots,x_n$ . Let $s,t$ be the generic structures of the species $\Sigma,\Theta$ respectively. Suppose that the following conditions are satisfied:

(1) We have a procedure of deduction $P\{x_1,\ldots,x_n,s\}$ of a structure of species $\Theta$ on $x_1,\ldots,x_n$ from a structure of species $\Sigma$ on $x_1,\ldots,x_n$ .

(2) We have a procedure of deduction $Q\{x_1,\ldots,x_n,t\}$ of a structure of species $\Sigma$ on $x_1,\ldots,x_n$ from a structure of species $\Theta$ on $x_1,\ldots,x_n$ .

(3) The relation $Q\{x_1,\ldots,x_n,P\{x_1,\ldots,x_n,s\}\}=s$ is a theorem in $\mathscr{T}_{\Sigma}$ , and the relation $P\{x_1,\ldots,x_n,Q\{x_1,\ldots,x_n,t\}\}=t$ is a theorem in $\mathscr{T}_{\Theta}$ .

The species of structures $\Sigma$ and $\Theta$ are then said to be equivalent by means of the procedures of deduction $P$ and $Q$ . In this case, for every theorem $B\{x_1,\ldots,x_n,s\}$ in the theory $\mathscr{T}_{\Sigma}$ , the relation $B\{x_1,\ldots,x_n,Q\}$ is a theorem in $\mathscr{T}_{\Theta}$ ; and conversely, for every theorem $C\{x_1,\ldots,x_n,t\}$ in the theory $\mathscr{T}_{\Theta}$ , the relation $C\{x_1,\ldots,x_n,P\}$ is a theorem in $\mathscr{T}_{\Sigma}$ .

☡

If $U$ is a structure of species $\Sigma$ , the structure deduced from $U$ by the procedure $P$ is said to be equivalent to $U$ . Criterion CST6 implies the following :

CST7. Let $\mathscr{S},\mathscr{S}'$ be two structures of species $\Sigma$ on the principal base sets $(E_1,\ldots,E_n),(E'_1,\ldots,E'_n)$ , respectively. Let $\mathscr{S}_0,\mathscr{S}'_0$ be structures of species $\Theta$ which are equivalent respectively to $\mathscr{S}$ and $\mathscr{S}'$ . In order that $(g_1,\ldots,g_n)$ should be an isomorphism with respect to the structures $\mathscr{S}_0$ and $\mathscr{S}'_0$ , it is necessary and sufficient that $(g_1,\ldots,g_n)$ should be an isomorphism with respect to the structures $\mathscr{S}$ and $\mathscr{S}'$ .

In practice, we make no distinction between the theories $\mathscr{T}_{\Sigma}$ and $\mathscr{T}_{\Theta}$ of two equivalent species of structures.

Examples

Example 1 03VI

\* Let $\Sigma$ be the species of commutative group structures; $\Sigma$ has a single (principal) base set $A$ , and its generic structure consists of a single letter $F$ ; the typical characterization of $\Sigma$ is $F\in\mathfrak{P}((A\times A)\times A)$ , and we denote the axiom of $\Sigma$ by $R\{A,F\}$ . This axiom implies in particular that $F$ is the graph of a function (the “law of composition” of the group; cf. no. 4, Example 2). In the theory $\mathscr{T}_{\Sigma}$ (where $\mathscr{T}$ denotes the theory of sets) we define a term $M\{A,F\}$ which is a functional graph in $\mathfrak{P}((\mathbf{Z}\times A)\times A)$ and satisfies the following relation $B\{M,A,F\}$ :

(\forall x)(\forall y)(\forall n)((x\in A\ \text{and}\ y\in A\ \text{and}\ n\in\mathbf{Z}) \Rightarrow (M(n,F(x,y))=F(M(n,x),M(n,y))))

and

(\forall x)(\forall m)(\forall n)((x\in A\ \text{and}\ m\in\mathbf{Z}\ \text{and}\ n\in\mathbf{Z}) \Rightarrow (M(m+n,x)=F(M(m,x),M(n,x))))

and

(\forall x)(\forall m)(\forall n)((x\in A\ \text{and}\ m\in\mathbf{Z}\ \text{and}\ n\in\mathbf{Z}) \Rightarrow (M(m,M(n,x))=M(mn,x)))

and

(\forall x)((x\in A)\Rightarrow (M(1,x)=x)).

(“multiplication of an element of $A$ by an integer”).

Consider the species $\Theta$ of $\mathbf{Z}$ -module structures, which has a single principal base set $A$ , with $\mathbf{Z}$ as auxiliary set, and whose generic structure contains two letters G, L, with the typical characterization

$G \in \mathfrak{P}((A \times A) \times A)$ and $L \in \mathfrak{P}((\mathbf{Z} \times A) \times A)$

and the axiom

“ $R \mid A, G$ and $(L$ is a functional graph) and $B \mid L, A, G$ ”.

It is immediately verified that the terms F, M constitute a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$ , and that the term G is a procedure of deduction of a structure of species $\Sigma$ from a structure of species $\Theta$ . Furthermore, the condition (3) above is trivially satisfied. We may therefore say that the species of commutative group structures and the species of $\mathbf{Z}$ -module structures are equivalent.

Example 2 03VJ

Let $\Sigma$ be the species of topological structures (no. 4, Example 3), let $A$ be the (principal) base set, and let $V$ be the generic structure of $\Sigma$ . Consider the relation

x \in A \text{ and } X \subset A \text{ and } (\forall U)((U \in V \text{ and } x \in U) \Rightarrow (X \cap U \neq \varnothing)).

This relation has a graph $P \subset \mathfrak{P}(A) \times A$ with respect to the pair $(X, x)$ ; $P \mid A, V$ is a term called “the set of all pairs $(X, x)$ such that $x$ lies in the closure of $X$ with respect to the topology $V$ ”. We can then prove (cf. General Topology, Chapter I, § 1) that the following relations are theorems in $\mathscr{T}_{\Sigma}$ :

P(\varnothing)=\varnothing,

(\forall Y)((Y \subset A) \Rightarrow (\forall Y \subset P(Y))),

(\forall Y)((Y \subset A) \Rightarrow (P(P(Y))=P(Y))),

(\forall Y)(\forall Z)((Y \subset A \text{ and } Z \subset A) \Rightarrow (P(Y \cup Z)=P(Y)\cup P(Z))).

Consider the species of structures $\Theta$ , having a single (principal) base set $A$ , whose generic structure consists of a single letter $W$ , which has as typical characterization $W \in \mathfrak{P}(\mathfrak{P}(A) \times A)$ and as axiom

W(\varnothing)=\varnothing \text{ and } (\forall Y)((Y \subset A) \Rightarrow (Y \subset W(Y)))

and

(\forall Y)((Y \subset A) \Rightarrow (W(W(Y))=W(Y)))

and

(\forall Y)(\forall Z)((Y \subset A \text{ and } Z \subset A) \Rightarrow (W(Y \cup Z)=W(Y)\cup W(Z))).

Consider also the relation

U \subset A \text{ and } (\forall x)((x \in U) \Rightarrow x \notin W(A-U)).

The set of all $U \in \mathfrak{P}(A)$ which satisfy this relation is a subset $Q \mid A, W$ of $\mathfrak{P}(A)$ . We can then prove (General Topology, Chapter I, § 1, Exercise 10) that the following relations are theorems in $\mathscr{T}_{\Theta}$ :

A \in Q,

(\forall M)((M \subset Q) \Rightarrow ((\bigcup_{X \in M} X) \in Q)),

(\forall X)(\forall Y)((X \in Q \text{ and } Y \in Q) \Rightarrow ((X \cap Y) \in Q)).

Thus the terms $P\{A,V\}$ and $Q\{A,W\}$ satisfy conditions (1) and (2) above, and it is easily seen that they also satisfy condition (3). The species of structures $\Sigma$ and $\Theta$ are therefore equivalent, and we therefore consider every structure of species $\Theta$ as a topology, namely that which corresponds to it under the procedure of deduction $Q\{A,W\}$ . \

Exercises

See the exercises for § 1.

[^1]: We use the notion of integer in the same manner as in Chapter I, that is to say, in the metamathematical sense of marks arranged in a certain order; this use has nothing to do with the mathematical theory of integers which was developed in Chapter III. [^2]: For typographical reasons we write here $f^{-1}$ instead of $\overset{-1}{f}$ .