Order Theory, Fixed Point, and Social Change

Daniel Parrochia

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# Order Theory, Fixed Point, and Social Change

Daniel Parrochia

University of Lyon (France)

> "Se vogliamo che tutto rimanga com'è, bisogna che tutto cambi"
> (If we want nothing to change, then everything must change).

Tancredi Falconeri, *Il Gattopardo* (The Leopard) (Luchino Visconti).

## Abstract.

Knowing that a society can be represented, in a simplified way, not only as a hypergraph but also as a partial order, we seek to draw some philosophical consequences from certain fixed-point theorems existing in order theory. Then, in the case of an established order that has taken the form of a set of controls (as in what modernity has called "control societies"), we show under what conditions such an order can collapse.

Keywords: Social hypergraph, Partial order, Lattice, Complete lattices, Fixed point, Idempotent dioid.

# 1 Societies and Partial Orders

## 1.1 The Social Hypergraph

We will start with an ultra-simplified "toy model" of society.

What, in fact, is a society when we consider essentially the subsets of which it is composed?

From a mathematical point of view, this is a set $S$ whose elements ($s_i$) are people, and whose various groupings, associations, or classes ($P_K$) form a set of subsets ($\mathcal{P}(S)$), with the following properties:
i) A subset $P_i$ different from $S$ can always be included in another. We can always write:

$$ \exists i, j, i \neq j \in I \mid P_i \subset P_j; $$

ii) In general, the intersection of two subsets is not empty, that is:

$$ \exists i, j, i \neq j \mid P_i \cap P_j \neq \emptyset. $$

The structure $H = \langle S, \mathcal{P} \rangle$ constitutes what is called a hypergraph<fnref n="1" />.

Example: $S$ = the set of inhabitants of the same country; $P_1$ = the class of farmers; $P_2$ = the class of industrialists; $P_3$ = the class of academics; $P_4$ = the class of tennis players; $P_5$ = the class of musicians, etc.

As one might expect, some of the elements $s_i$ can belong to more than one class.

Therefore, in a society obviously simplified to the extreme, there exists a relatively accessible algebra of relations that constitutes an important chapter of "social statistics."

## 1.2 Society and Partial Orders

One can sometimes identify society, or some of its parts, with a slightly stronger structure. It is often possible to define what is called a "partial order" on social subsets, with particular properties.

The works of mathematicians<fnref n="2" /> have led to the emergence of a terminology that can

<footnote n="1">¹The notion of a social hypergraph has appeared for a very long time in numerous publications (see for example [Chapelle (de la) 20]; or already [Bae-Kim 09]). In general, the application of mathematics to the social sciences, developed in France by M. Barbut (in Paris) or M. Brissaud (in Lyon) has promoted the use of this kind of structure.</footnote>

<footnote n="2">²Historically, interest in the notion of order in mathematics dates back to the beginning of the 20th century with Peirce (1880) and Schröder (1890). Among the first specific works on partial orders, we find an article by MacNeille (1937) and then the classic work by Birkhoff (1948) on the notion of lattice. A little later, Kurepa (1935) worked on this particular class of partial orders, namely trees. The theory of orders was subsequently developed by Hausdorf (1949), while Dushnik and Miller (1941) had already developed a reflection on the dimension of ordered sets. At the end of the 1950s came the first exclusive contribution to a general theory of ordered sets, that of the Spanish mathematician N. Cuesta Dutari (1958). But it wasn't until the 1980s that important books on the subject appeared, such as those by Erné, Fraïssé, and Fishburn, as well as a journal</footnote>
be summarized as follows:

**Definition 1.1.** A relation $R$, defined on a set $E$, equips $E$ with a structure:

- of quasi-order (or pre-order) if it is reflexive and transitive;

- of strict order if it is irreflexive and transitive;

- of order (pure and simple) if it is reflexive, antisymmetric, and transitive;

- of linear order (or total order) if $R$ is an order relation, and if, in addition, for every element $a \neq B$, either $aRb$ or $bRa$.

We denote by $(E, \leq)$ a set equipped with an order relation.

**Definition 1.2 (Partial Order).** If elements of the ordered set $(E, \leq)$ are incompatible, the order is said to be "partial" and the set $(E, \leq)$ is said to be a partially ordered set (or *poset* for short) (see an example in Fig. 1).

<figure><img src="image_2.png" /><figcaption>Figure 1: A partial ordered set</figcaption></figure>

A society can, of course, be seen as a partial order, but this order is not necessarily immutable. One of the major sociological problems, that of social change, can then be approached in several ways.

The first of these is to ask whether it is possible for something to remain despite changes (which can be radical), and this question is, mathematically, that of finding one (or more) fixed point(s) in the underlying set. Let us therefore begin by formally defining this notion.

<footnote>(ORDER) entirely devoted to the question of order. From the 1990s onward, the theory has been fully established as such in mathematics. For this history and the associated bibliography, see ([Parrochia 23], 74-75, [Parrochia-Neuville 13]).</footnote>
## 2 The Notion of a Fixed Point

The notion of a "fixed point," whose original domain of definition is Analysis, is
defined as follows:

**Definition 2.1.** Let $f$ be a function defined and continuous on a stable interval $I$. A fixed point is any $x \in I$ such that $f(x) = x$. If $f$ is defined on an interval $I$ of $\mathbb{R}$, this property translates graphically to the fact that the graph of $f$ intersects the line $y = x$, the point of intersection being precisely the "fixed point" in question.

This situation was generalized by Brouwer, who obtained the following theorem:

**Theorem 2.1** ([Brouwer 12]). Every continuous function from a non-empty compact convex set of $\mathbb{R}^n$ to itself has a fixed point.

One consequence is that, regardless of the continuous agitation, perturbations, or
transformations applied to a sufficiently homogeneous domain (fluid eddies, crum-
pling of a sheet of paper, etc.), there is at least one element that will always be in
the same place.

Over time, many variations of fixed-point theorems have been discovered. Among the
hundred or so that currently exist, we would like to focus on three that are directly
relevant to order theory and, therefore, can be applied to "social order."

## 3 Three Fixed-Point Theorems in Order Theory

First, in order theory, there is a fixed-point theorem that holds for complete partial orders. Let's begin by defining this concept.

**Definition 3.1.** A *complete partial order* is a partially ordered set that has a minimum element, and all of whose paths have a least upper bound.

### 3.1 Kleene's Theorem

Let us now introduce the notion of a Scott continuous function, which we will need
to state Kleene's fixed-point theorem.

**Definition 3.2.** *f* is a *Scott continuous function* if it is an increasing function that also preserves the least upper bounds of the paths. (The fact that it is increasing ensures *a priori* that it has a smallest fixed point, and that the sequence discussed in the theorem is increasing.)
We can now move on to the statement of the theorem:

**Theorem 3.1.** Let $L$ be a complete partial order, 0 its minimum element, and $f: L \to L$ a continuous Scott function. Then the smallest fixed point of $f$ is the supremum of the following increasing sequence:

$$0 \le f(0) \le f^2(0) \le \dots \le f^n(0) \le \dots$$

## 3.2 The Bourbaki-Kneser Theorem

To understand the essence of the second fixed-point theorem, let us first define the notion of an inductive or a strictly inductive set:

**Definition 3.3.** A set $E$, equipped with an order relation $\le$ and such that each of its paths (totally ordered parts) has a least upper bound, is called an *inductive set* or a *strictly inductive set*. A strictly inductive set necessarily has a least upper bound, the least upper bound of the empty path.

With this established, the theorem states:

**Theorem 3.2.** Let $f$ be a function from a strictly inductive set $(E, \le)$ to itself that is expansive, that is, satisfying $x \le f(x)$ for all $x$ in $E$. Then $f$ has at least one fixed point, that is, an element $m$ of $E$ satisfying $f(m) = m$.

This theorem, initially attributed to Bourbaki but stated without proof in a 1939 booklet, originally considered inductive sets to be partially ordered sets whose chains possess a least upper bound. Later editions simply used an upper bound (see [Kanamori 97], 303, note 45). It is now associated with Kneser's name because, in 1950, he proved a version of Zorn's theorem for well-ordered chains, which is a variation in the definition of the notion of an inductive set. In his article, Kneser cites Zermelo's work (1904) as a source of inspiration (see [Kneser 50]).

We have thus far assumed that the set on which we are working is a partially ordered set (or poset). If, in addition, this poset is such that every pair of elements in it has the same infimum and the same supremum, then we are dealing with what is called a lattice.

We will now clarify this definition and recall some properties of this notion that will be applied later.

## 3.3 The Knaster-Tarski Theorem

In what follows, $(E, \le)$ will always denote a (non-empty) partially ordered set.
**Definition 3.4.** A lattice is a partially ordered set $(E, \leq)$ in which every pair $\{x, y\}$ of $E$ has the same infimum (greatest lower bound) and the same supremum (least upper bound).

**Definition 3.5.** A complete lattice is a partially ordered set $(E, \leq)$ in which every non-empty subset $D$ of $E$ has the same infimum and supremum.

We then have the following theorem:

**Theorem 3.3 (Knaster-Tarski, 1928, 1955).** Let $(E, \leq)$ be a complete lattice. Every increasing function $f$ from $E$ to $E$ has at least one fixed point. The set $\text{Fix}(f)$ of fixed points of $f$ is a complete lattice for the induced order.

*Proof.* We proceed in three steps:

**Step 1:** Since $E$ is a complete lattice, $E$ has a lower bound and a supremum, which are its smallest and largest elements. We denote them respectively $\alpha$ and $\beta$. The set $A = \{z \in E, f(z) \leq z\}$ is non-empty because $\beta \in A$. Let $a$ be its lower bound. Similarly, the set $B = \{z \in E, f(z) \geq z\}$ is non-empty because $\alpha \in \beta$. Let $b$ be its supremum. If $z \in A$, then $a \leq z$ and, by the increasing nature of $f$, we have $f(a) \leq f(z) \leq z$ (since $z \in A$). We deduce that $f(a)$ is a lower bound of $A$ and therefore that $f(a) \leq a$. But then, by increasing $f$, we have $f(f(a)) \leq a$, and therefore, $f(a) \in A$, hence $a \leq f(a)$; since $a$ is the infimum of $A$.

We have shown that $a \leq f(a)$ and that $f(a) \leq a$, and therefore $f(a) = a$. Similarly, we show that $f(b) = b$.

**Step 2:** It remains to show that $\text{Fix}(f)$ is a complete lattice. If $c \in \text{Fix}(f)$ then $c \in A \cap B$ and therefore $a \leq c \leq b$, thus $\text{Fix}(f)$ has a smallest and a largest element.

**Step 3:** Let $(c_i)_{i \in I}$ be a family of fixed points of $f$, we denote $c := \text{inf}\{c_i$ (note: here, $c \in E$, but nothing says that it is a fixed point of $f$). Then, for all $i \in I$, we have $f(c) \leq f(c_i) = c_i$. We have shown that $f(c)$ is a lower bound for $(c_i)_{i \in I}$, and therefore; $f(c) \leq c$. Let's introduce the set $S := \{z \in E, \alpha \leq z \leq c\}$. Since $f$ is increasing and $f(c) \leq c$, then $S$ is stable under $f$ and $S$ is a complete lattice for the induced order. Applying steps 1 and 2, we deduce that $f$ admits a major fixed point on $S$, which is therefore a lower bound of the $(c_i)_{i \in I}$. We proceed similarly to show that the family of $(c_i)_{i \in I}$ has a least upper bound. Consequently, $\text{Fix}(f)$ is a complete lattice for the induced order. $\square$
**Note:** There is a converse to this theorem: if every increasing function from $E$ to $E$ has a fixed point, then $E$ is a complete lattice.

In this case, the two sequences $(x_n)$ and $(y_n)$ defined by $x_{n+1} = f(x_n)$ and $y_{n+1} = f(y_n)$ satisfy: there exists $(x, y) \in E \times E$ such that $x_n \uparrow x, y_n \downarrow y, f(x) = x, f(y) = y$ and $x \le y$.

Furthermore, if $a$ is a fixed point of $f$ such that $a \ge x_0$, then $a \ge x$. Similarly, if $b$ is a fixed point of $f$ such that $b \le y_0$, then $b \le y$.

## 3.4 Examples of Incomplete Lattices

By definition of a complete lattice, every finite lattice is complete. Therefore, every increasing function in a finite lattice necessarily has at least one fixed point, and there may even be several, in which case they are ordered relative to each other to form a lattice themselves.

On the other hand, if $(E, \le)$ is not a complete lattice, which in fact implies that it is an infinite lattice, the same does not hold true. This kind of set, containing either a completely unordered subset or an infinite chain (a linear order), cannot be complete.

**Example 1:** The set $(\mathbb{R}, \le)$ is a lattice, but it is not complete because it has no largest element and/or $[0, +\infty[$ has no least upper bound. The function $f: x \in \mathbb{R} \mapsto x - 1 \in \mathbb{R}$ is increasing but has no fixed point.

**Example 2:** Similarly, the set $\mathbb{Q}$ of all rational numbers, equipped with the usual linear order, is an infinite distributive lattice that is not complete. For example, $\mathbb{Q}$ itself has neither a least upper bound nor a greatest lower bound.

**Example 3:** Similarly again, the non-empty subset of $\mathbb{Q}$ such that $\{x \in \mathbb{Q} : x^2 < 2\}$.

**Example 4:** Now consider the lattice of finite subsets of $\mathbb{N}$. Since the union and intersection of any two finite subsets is also finite, it is certainly a lattice. But the set $A = \{0, \{0, 1\}, \{0, 1, 2\}, \dots\}$ has no least upper bound in the lattice. Naturally, $\mathbb{N}$ is a least upper bound of $A$ in $P(\mathbb{N})$, the power set of $\mathbb{N}$.
## 3.5 Philosophical Interpretation

Therefore, there is no fixed point for increasing functions in an infinite partially ordered set. Philosophically speaking, it would obviously be interesting to know, from this point onward, in what domain (finite or infinite), but which we can assume to be "partially ordered," we live...

This leads us back to Pascal’s questions, sensitive not only to the relativity of movement but also to the universal mobility of things, even if his use of the word "fixed point" does not have the formal character we have attributed to it so far:

> "Those who are in disorder tell those who are in order that it is they who are straying from nature and they believe they are following it, just as those who are on a ship believe that those on the shore are fleeing. Language is the same on all sides; one must have a fixed point to judge it. The port judges those who are on a ship, but where shall we find a port in morality?" (see [Pascal 93], 383).

Or again:

> "When everything is in motion equally, nothing appears to be moving, as in a ship: when all are heading towards overflowing, no one seems to be. The one who stops makes the others’ impetuosity noticeable, like a fixed point" (see [Pascal 93], 382).

That being said, and failing to find a fixed point in morality, there is a second way to pose the question of social change. It is to ask under what conditions, for example in the context of a surveillance society or a control society, the order on which these are founded could collapse completely, without anything, or very little, remaining. To conduct such a study, we will examine the relationships between chains and antichains in a partially ordered set and, more specifically, in a lattice.

# 4 Antichains of a Lattice and Lattice of Antichains

**Definition 4.1** (Antichain). An antichain A is a subset of a partially ordered set E whose elements are pairwise incomparable.

The concept is so named in contrast to chains, which are subsets of an ordered set and whose elements are always pairwise comparable (see [Caspard et al.]).

**Definition 4.2** (Maximum Antichain). An antichain is said to be maximal if it is not contained in any other antichain in the set.
**Theorem 4.1.** Let P be a finite poset. The set $\mathcal{A}(P)$ of all antichains of P equipped with the union and intersection operations is a distributive lattice.

We will not give the (easy, but somewhat lengthy) proof of this theorem, which follows from a result of Birkhoff (the representation theorem for finite distributive lattices) according to which every finite distributive lattice is isomorphic to the lattice of ideals of a finite poset ordered by inclusion (see [Birkhoff 33], [Birkhoff 37], [Birkhoff 40]). The fact that the ideals are nothing other than maximal antichains became clearer, however, from the work of Sperner (see [Sperner 28]) and Dilworth (see [Dilworth 50-51]). It is only from [Birkhoff 67], [Grätzer 78], [Stanley 86] that a combinatorial reformulation of this result, showing the ideals as downsets<fnref n="3" /> and the antichains as canonical representatives, allows the theorem to be stated in the form we have given it.

Let P be a finite poset. The set $\mathcal{MA}(P)$ of maximal antichains of P equipped with the "dominance" relation is a (non-distributive) lattice. In formulas:

if we define:

$$A \leq B \iff \forall a \in A, \exists b \in B \text{ such that } a \leq b,$$

then, $(\mathcal{MA}(P), \leq)$ is a lattice.

It was Dilworth, in 1950 (see [Dilworth 50-51]), who introduced the deep structural analysis of maximal antichains and implicitly used the dominance order. However, he did not explicitly state that maximal antichains form a lattice. It wasn't until 1978 and an article by Duffus and Rival (see [Duffus-Rival 78]) that a systematic study of maximal antichains and an explicit formulation of the theorem were undertaken.

We will not provide a proof of this theorem here, but will only offer an illustration. Fig. 2 represents the poset P, the distributive lattice of its antichains $A(P)$, and finally, the non-distributive lattice of its maximal antichains $\mathcal{MA}(P)$.

# 5 A Model for a Control Society

According to some philosophers of the last century (Michel Foucault or Gilles Deleuze, for example<fnref n="4" /> our society is a "control society". But a control society is not a soci-

<footnote n="3">³A subset D of a poset P is a downset if $x \in D, y \leq x \Rightarrow y \in D$.</footnote>

<footnote n="4">⁴It is generally agreed that the germ of the concept of "control society" can be found in the writings of Michel Foucault, notably ([Foucault 75], 211-213, [Foucault 01¹], 1299, or [Foucault 01²], 532-533). In reality, it seems that Gilles Deleuze, borrowing this phrase from the novelist W. S.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2: Poset, Antichain Lattice, and Maximal Antichain Lattice</figcaption></figure>

ety that is controlled, nor even a society where control is exercised; it is a society where control is exerted primarily on individuals indirectly, that is to say, not first through the training and marking of bodies (as in so-called "disciplinary societies") but through suggestions, humiliations, propaganda, ideologies, disinformation, noise, and various forms of interference, and only secondarily through coercion.

Nevertheless, this society, given its injustices and inequalities, which are always liable to generate sedition, remains a society of mistrust, and still knows and practices threats and sanctions. This implies that it still exercises controls which, even if less violent than before and, for economic reasons, only intermittent, are supposed to monitor the behavior of individuals, including those who control others. In this surveillance society, control therefore has exponents<fnref n="5" />.

Burroughs, essentially developed it, following in Foucault's footsteps, on several occasions: first, rather briefly in his book on Foucault (see [Deleuze 86]); then in a lecture given on May 17, 1987, at La Fémis on the notion of the act of creation; also during an interview in the spring of 1990 with the Italian philosopher Antonio Negri, for the first issue of the journal *Futur Antérieur*, and finally in "Postscript on control societies", written in May 1990 for the first issue of *L'Autre journal* published around the same time, echoing the interview with Negri, these two latter writings being reprinted in the collection entitled *Pourparlers 1972-1990* (see [Deleuze 03]). Hardt and Negri would then have followed suit (see [?]).

<footnote n="5">⁵Bachelardian innocence attributed these same exponents to a kind of intimate surveillance, understood as "intellectual self-surveillance," the quintessential scientific attitude which was to culminate – sweet reverie – in the poetic attitude (see [Bachelard 70], 65-81).</footnote>
## 5.1 Mathematical Idealization

We therefore begin by defining (toy model) the existence of a fictitious entity $S = s_1 \times s_2 \times \dots \times s_m$, which we will call, to be consistent with what we said earlier, a "society of mistrust" or a "surveillance society," and which has the property of being decomposable into a finite number of strata $s_j$, which are all dimensions of social life (work, leisure, travel, economic behaviors, beliefs, etc.).<fnref n="6" />

As we stated earlier, in a society of mistrust or surveillance, control has exponents. If $C$ is a control (we don't need to specify its form), $C^2 = C * C$ will be the control of the control, $C^3 = C * C^2$ the control of the control of the control, and so on. To avoid redundancy in these controls, we will first assume that the operation *, interpreted as ordinary multiplication, is not idempotent. Furthermore, to give these same controls an appearance of objectivity, it is necessary to avoid any circuit<fnref n="7" />.

Since the same individual can be subject to different controls, we also define a second law, denoted $⊕$, such that the sum $C⊕C'$ is also a control. For simplicity, let's identify every control with its result, a polynomial with Boolean variables, which will take the value 1 if the control is good, and the value 0 otherwise.

The laws $*$ and $⊕$ are associative and commutative, with $\epsilon = 0$ being the identity element of $⊕$ while $e = 1$ is the identity element of $*$.

By interpreting $⊕$ as the symmetric difference, that is, by setting $C ⊕ C' = C + C' - CC'$, we note that this law is idempotent (since $C$ can only take the values $\ddot{A}\tilde{a}$, $\ddot{A}\tilde{a}1$ or 0, $C ⊕ C = C$). As we also verify the distributivity of $*$ with respect to $⊕$, the structure $(S, *, ⊕)$ is that of a dioid (or semiring) (see Gon2).

We can then associate order relations with these laws. Let us consider, for example, the law $⊕$: we associate with it the following partial order relation (reflexive, antisymmetric, transitive): $a \ge b \iff \exists c \in S \text{ tel que } a = b \oplus c$.

A number of consequences follow from the model, as we have defined it.

<footnote n="6">⁶In what follows, we will be adapting elements from an article published in [Parrochia & Tirloni 12], 211-218).</footnote>

<footnote n="7">⁷Let's be clear: we are not denying the existence of local control loops such as, for example, $C^1$ controls $C^2$ which controls $C^3$ which controls $C^1$, and so on. But we are excising them because, from the point of view of the overall structure, they behave like a single control $C$ which continues to be taken in a partial order. And it should not be implied that the entire system is closed-loop. The control can be "diffuse" or "invasive" (network-wide) or whatever you want. Ultimately, there is always something uncontrolled, that is, a final element for each chain.</footnote>
**Proposition 5.1.** Each stratum of society S is, relative to the control procedures applied to it, a partially ordered set.

Let $(s_1, \ge 1), (s_2, \ge 2), \dots, (s_m, \ge m), m$ be strata of S, which we assume, each time, to be partially ordered by a control relation $\ge_j$. $S = s_1 \times s_2 \times \dots \times s_m$ being the Cartesian product of the underlying sets, we define a binary relation $\ge$ on S, such that:

$$ (C_1, C_2, \dots, C_n) \ge (C'_1, C'_2, \dots, C'_n) \iff C_1 \ge C'_1, C_2 \ge C'_2, \dots, C_n \ge C'_n. $$

In this case, we know that:

**Proposition 5.2.** The relation $\ge$ defined on the product of the m partially ordered strata $(s_1, \ge 1), (s_2, \ge 2), \dots, (s_m, \ge m)$ is itself a partial order on S, and this order, denoted $(S, \ge)$, is called a "Cartesian" order.

Since the control instances, wherever they are exercised, are always finite in number, there is necessarily, on each stratum of S, a number k such that $C_k = C_1 \oplus C_2 \oplus \dots \oplus C_{k-1}$ is the last state of control. A control society is therefore a society where control exists everywhere, but where, also, on each stratum $s_j$ where a control relation has been defined, every partial order has an upper bound, and thus every control chain has a final element, easily identifiable, moreover, on the Hasse diagram of partial order, where the maximal chains (total, antiflexive, antisymmetric, and transitive orders) are precisely located. This shows that any surveillance society is a society that constantly creates uncontrolled elements.

It should also be noted that, in such a society, since indirect control can only be exercised approximately, and since individual behaviors partially escape control chains, the tendency is obviously to densify the chains and, consequently, to multiply them.

Let us now recall the formal definitions of an antichain and a maximal antichain:

Given P a set equipped with an order relation $\le$, a subset A is an antichain of P if for all $x, y$ in A,

$$ x \le y \text{ or } y \le x \implies x = y. $$

An antichain A is maximal if $A \not\subseteq B, \forall B \in P$. A well-known theorem then shows that:

**Theorem 5.3 ([Kurepa 52])**. Every (partially) ordered set contains a maximal antichain.
This theorem is a consequence of a result by Vaught, which asserts the existence of a maximal disjunctive family in any ordered set ($F, \subseteq$) equipped with the inclusion-equality relation. It should be noted that Vaught's result, like [Birkhoff 48] (existence of a maximal chain in any ordered set) or Zorn's lemma (if ($P, \leq$) is a partially ordered non-empty set such that every chain in $P$ has a least upper bound, then $P$ has a maximal element), is in fact equivalent to the axiom of choice (every family of non-empty sets $F$ has a choice function $f$ such that, for $X$ in $F$, $f(X)$ belongs to $X$).

We know that such a set is postulated without being defined (and sometimes it is even undefinable). Such an axiom is therefore not insignificant. It amounts to saying that, for any non-empty set $X$ of the family $F$ under consideration, there exists a set whose intersection with each element of $X$ is a set with only one element (a singleton).

The axiom of choice thus implicitly presupposes that every singleton is in fact potentially part of a set: an anti-Stirnerian axiom<fnref n="8" />, if one may put it that way, and therefore anti-anarchist.

Accepting Kurepa's result, and therefore Vaught's (which it presupposes), thus leads to accepting the axiom of choice (which is, moreover, provable for finite sets).

Under this assumption, we then realize that such antichains can easily be found in a control society. It suffices to partition the set of controls into levels $N_k$ and define an ordinal function or rank function $r(X_i)$ such that $X_i$ belongs to $N_k$ and implies $r(X_i) = k$. In particular, any set of uncontrolled elements in a stratum of $S$ constitutes precisely such an antichain.

Note that this property of being an antichain is passed on to the product set:

**Proposition 5.4.** If $A_1, A_2, ..., A_p$ are antichains, their product $A_1\text{flfl} \times A_2 \times ... \times A_p$ is also an antichain.

A particularly salient fact concerning products of partial orders is the following:

**Proposition 5.5.** The product will never be a chain unless one of the partial orders

<footnote n="8">⁸The philosopher Max Stirner (1806-1856), with *The Unique and its Property (Der Einzige und sein Eigentum, 1844)*, wrote a radical indictment against all the higher powers from which the individual alienates his "I" (State, religion, society, humanity). He urges everyone to appropriate what is in their power, independently of the various forces of oppression external to the individual, located at the center of everything, which makes him, according to John Henry Mackay, the father of individualist anarchism.</footnote>
is trivial, that is, a singleton.

*Proof.* Consider a product of two antichains $A = A_1 \times A_2$, with $(a,b), (c,d)$ belonging to $A$. Suppose then that $A$ is a chain. Since $(a,b)$ and $(c,d)$ belong to $A$, they are comparable, that is, $(a,b) \geq (c,d)$, which implies that $a \geq_1 c$ and $b \geq_2 d$. Similarly, $(c,b)$ and $(a,d)$ are also comparable. But since $b \geq_2 d$, we must have $(c,b) \geq (a,d)$, which means that $c \geq_1 a$, in other words, $a = c$, or $A_1 = \{a\}$. $\square$

One way to prevent society $S$, identified with the product of its strata, from becoming a total order is, paradoxically, to proliferate controls on each stratum.

Let us then rely on this very powerful property of the set of antichains that we have already encountered and which is expressed in Theorem 4.2, namely that the set of maximal antichains of a finite partially ordered set is a lattice.

Let us call such a lattice $T$, which is also well characterized because it is isomorphic to the lattice of ideals of the set. It immediately becomes clear that the upper bound of the lattice $T$ of antichains constitutes the largest subset of totally uncontrolled elements. By definition, a set of elements that escape all control tends to act independently. It is, at least virtually, "subversive." Let us therefore call such an entity a "potentially subversive subset." From this, we deduce Property 5:

**Proposition 5.6.** Every control (or surveillance) society contains at least one potentially subversive subset.

In such a context, one might think that the project of overthrowing a control society should consist of increasing the cardinality of the upper bound of the lattice $T$ of maximal antichains as much as possible. However, order theory shows the existence of a direct link between the size of an antichain and the number of chains in a partially ordered set:

**Theorem 5.7 ([Dilworth 50-51])**. In any partially ordered set $S$, the maximum size of an antichain is equal to the minimum number of chains in the chain decomposition of $S$.

Two consequences follow from this theorem: 1) Control and uncontrollability go hand in hand, and it is illusory, in a control society, to hope to annihilate only one of the two poles; 2) However, it is desirable to reduce the number $p$ of control chains in $S$ to this minimum $n(C)$, which corresponds to the size of the largest uncontrolled antichain. Sperner's theorem (see [Sperner 28], [Anderson 87], 2) gives the upper
bound $n(C) = n!/(n/2)!(n/2)!$ for a set of $n$ elements. Any society where $p > n(C)$ is necessarily an unnecessarily fussy society.

## 5.2 An inevitable revolution

Let us now consider the conditions under which a control society such as $S$ can be dismantled. Let us return to our initial dioid $(S, *, \oplus)$, with the operation $\oplus$ as the symmetric difference $(a \oplus b = a + b - ab)$, and the operation $*$ as ordinary multiplication. But let us now suppose that $*$ has been made idempotent, so that we have: $C^2 = C, C'^2 = C'$, etc. The idea is that the control of a control is nothing more than the control itself, which is a Spinozist-type postulate<fnref n="9" />. Therefore, it becomes pointless to proliferate controls. This already brings about the ruin of all our surface constructions, which are, moreover, perfectly artificial (holding companies, ministries, various papacies, rating agencies, internal affairs departments, doctoral schools, etc.). As the two laws are now idempotent, $(S, \oplus, *)$ is a double-idempotent dioid.

Since the operations $\oplus$ and $*$ are associative and commutative, we can also verify:

$$ \begin{aligned} (C \oplus C')^2 &= (C + C' - C \cdot C')^2 = C^2 + C'^2 + C^2 C' + 2CC' - 2C^2 C' - 2CC'^2 \\ &= C + C' + CC' + 2CC' - 2CC' - 2CC' = C + C' - CC' = C \oplus C'. \end{aligned} $$

$$ \begin{aligned} \text{- The distributivity of } *: C(C' \oplus C'') &= C(C' + C'' - C'C'') = CC' + CC'' - CC'C'' = \\ &CC'' + CC'' - CC' \cdot CC'' = C' \oplus CC''. \end{aligned} $$

## 5.3 The graph of controls

In a very general way, let us now represent a control as a test verifying the connection between certain pairs of points in a network: for example, does the individual or action $i$ satisfy criterion $j$? Any control can then be represented by an arc $(i, j)$ of a graph $G$, to which a Boolean variable $y_{ij}$ can be associated. The question of the reliability of social control now becomes that of determining the probability $p_{ij}$ of the existence of this arc. Assuming that these probabilities are independent, we want to

<footnote n="9">⁹In Spinoza (see [Spinoza 65], scholium to Proposition 21 of Book II of the *Ethics*), the idea of the idea "is nothing other than the form of the idea, insofar as the latter is considered as a mode of thinking without relation to the object; similarly, someone who knows something knows, by that very fact, that he knows it, and he knows at the same time that he knows that he knows, and so on ad infinitum."</footnote>
determine the probability that vertex $j$ is connected to vertex $i$ by a certain path.
We can then make the following remarks:

$B_{ij}^k$ will be a polynomial in $y$, ($B_{ij}^k = B_{ij}^k(y)$) such that $B_{ij}^k(p)$ represents the probability that $j$ is connected to $i$ by a path of length $k$ at most.

Since $\oplus$ is If $E$ is the identity matrix, we can define a matrix $B' = EflB$, and, for $0 \neq r \neq f \neq l \neq k$, $B^{r\kappa} = \sum B^r = B^k$.

$B_{ij}^{r\kappa}(p)$ then represents the probability that $j$ is connected to $i$ by a path of length $k$ at most.

Let $B^* = B^{r^{N-1}}$ be the transitive closure of the graph $G$. For any $C$ belonging to $S$, $eflC = 1+C-C=e$, so every circuit is 0-absorbing. We can then apply Gondran's theorem (see [Gondran 79], 78), which ensures that, if a graph $G$ is without 0-absorbing circuits, then $B^k$ has a limit $B^*$ as $k$ tends towards infinity, a limit that is reached for $k \le N-1$, when $N$ is the maximum number of vertices of an elementary path.

We therefore deduce from the preceding the actual existence of this limit $B^* = B^{r^{N-1}} = B^{(N-1)}$.

$B_{ij}^*$ and such that $B_{ij}^*(p)$ is the probability that $j$ is connected to $i$.

Hence:

**Proposition 5.8.** *S almost certainly collapses if* $B_{ij}^k(p) \ll B_{ij}^*(p)$.

In simpler terms: in a surveillance society, but where controls, although having exponents, obey two idempotent operations such as the $*$ and $\oplus$ operations described earlier, they could be represented, as explained above, by certain pairings of points in a graph, some representing individuals or actions, others criteria, the probability of these pairings defining the probability of existence of the graph's arcs. Gondran's theorem defines the condition that must be satisfied for these connections to exist, and therefore, for the graph to be connected or, if it is a network (for example, an energy transmission network), for it to be "reliable". Below this threshold, the condition of connectivity or reliability is not met, and the graph can break down into different components, connected or not, or even unravel if one is far below the threshold.

If we then identify this graph with the society $S$, it means that, in such a case, the society would necessarily decompose and collapse. It is quite possible that this
process has already begun in our societies, where hierarchical control structures are
becoming increasingly contested.

In France, for example, for the past fifty years, numerous controversies (the Creys-Malville breeder reactor, the Fessenheim nuclear power plant, the new Nantes airport, etc.) have called into question the decisions of the State, even though these decisions were validated by official control chains, each level of which is associated with a democratically elected representative of the people or an expert capable of conducting rational analyses. This means that these chains are collapsing ($C^n = C$) and the sole remaining authority, perceived as arbitrary as a whole, is violently attacked.

Furthermore, faced with the inequity of society, whose meritocratic systems no longer function (the selection process being largely rigged), illegal alternatives in the form of mafia-like networks are developing, particularly in the suburbs, giving rise to a kind of parallel state or administration. These two phenomena combined testify to the fragmentation of society.

# 6 Conclusion

As perhaps Visconti's adage, put in the mouth of Tancredi, the cynical aristocrat in "The Leopard", suggested, even when everything changes in a dynamic, evolving system subject to sufficiently smooth processes of transformation, at least one fixed point always remains, and sometimes even several.

On the other hand, a surveillance society identified with a partial order in which controls enter into idempotent operations changes the partial order into a double idempotent dioid, and if we consider a case where the reliability of tests or controls is far below the reliability limit, then there is a high chance that the society will break down and ultimately collapse.

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