A Jigsaw Puzzle

The article discusses a jigsaw puzzle featuring a portrait of Carl Friedrich Gauss in his old age, illustrating various states of the puzzle through Figures 1.1(a) to 1.1(d) A state is defined as any partial solution formed by a connected subset of correctly positioned puzzle pieces, with the completed puzzle and the initial empty board also considered states A total of 41 distinct states are identified Each of the six puzzle pieces can undergo two transformations: placing a piece on an empty board or linking it to existing pieces, and removing a piece while ensuring the remaining pieces remain connected These transformations are applied universally to all states, meaning that placing or removing a piece not currently on the board does not alter the state.

In this article, we introduce the concept of a 'medium', defined as a pair of sets consisting of a set of states (S) and a collection of transformations (T) that can convert one state into another The formal definition of this structure is based on two key axioms, which are detailed in Definition 2.2.1.

Figure 1.1.Four states of a medium represented by the jigsaw puzzle: Carl Friedrich Gauss in old age The full medium contains 41 states (see Figure 1.2).

The transformations discussed are not one-to-one, as demonstrated by the example of adding the upper left piece of the puzzle, which leads to the same state regardless of the initial configuration This creates a loop, indicating that the transformations associated with each piece are not mutual inverses However, each transformation can reverse the effect of the other, which we define as 'reverses' of one another For a formal definition of 'reverse' in a broader context, refer to section 2.1.1.

In Gauss's puzzle, the representation of a medium can be effectively illustrated through its graph when the number of states is finite Figure 1.2 below showcases the graph associated with Gauss's puzzle, with the standard practice of omitting loops.

The Gauss puzzle medium is illustrated in Figure 1.2, showcasing a graph with six numbered pieces (1 to 6) and 41 vertices, each representing a distinct state or partial solution of the puzzle Each state is depicted within a rectangle that lists the included pieces The edges of the graph denote pairs of mutually reverse transformations—one for adding a piece and the other for removing it To maintain clarity, only select edges are labeled with circles.

An examination of this graph leads to further insight For any two states

To transform state S into state T, one can identify a sequence of allowed transformations that connect the two states without deviating from the permitted set This transformation path can be optimized to its minimal length, which corresponds to the number of unique elements present in state T that are not found in state S Additionally, any two transformation paths from S to T will have the same length, ensuring consistency in the transformation process.

In Figure 1.2, we illustrate two distinct paths from state 34 to the completed puzzle, highlighting the different orders of the same transformations by coloring their edges in red and blue.

A medium can be defined using a standard jigsaw puzzle, following specific rules These media exhibit a unique property where their transformations can be divided into two equal classes: one for adding pieces and another for removing them This distinction, referred to as 'orientation,' highlights an asymmetry present in various contexts In this terminology, a transformation belongs to a particular class if its reverse is in the opposite class However, it is essential to note that there are significant instances where such a natural orientation is absent, indicating that this concept is not a fundamental aspect of defining a medium.

In fact, the next two examples involve media in which no natural orienta- tion of the set of transformations suggests itself.

A Geometrical Example

In a finite collection of hyperplanes in R^n, the space R^n minus the union of these hyperplanes forms a union of open, convex polyhedral regions These regions, which may be bounded or unbounded, are considered as distinct states within the arrangement.

In a finite collection of states, it is always possible to transition from one state to an adjacent one by crossing a hyperplane that includes a facet of the collection This process is formalized through the transformation of states, with the assumption that only one hyperplane is crossed at a time.

Each hyperplane \( H \) in \( A \) is associated with two ordered pairs, \( (H^-, H^+) \) and \( (H^+, H^-) \), representing the open half spaces separated by the hyperplane These pairs give rise to two transformations, \( \tau_{H^+} \) and \( \tau_{H^-} \), which modify states based on their position relative to the hyperplane The transformation \( \tau_{H^+} \) shifts a state to an adjacent state in \( H^+ \) when possible, while leaving it unchanged otherwise Conversely, \( \tau_{H^-} \) transitions a state to an adjacent state in \( H^- \) if feasible, or remains unchanged if not Formally, applying \( \tau_{H^+} \) to a state \( P \) yields another state \( Q \) when \( P \) is within \( H^- \) and \( Q \) falls within \( H^+ \).

In this case, the transformations of the two paths illustrated in (1.1) and (1.2) occur in reverse order; however, it is important to note that not all pairs of distinct paths exhibit this reversal (Problem 1.1).

In larger puzzles, such as the 3×3 configuration, it is common to encounter states with missing pieces, creating interconnected regions with holes The polyhedral regions P and Q share a facet that lies within the hyperplane separating H− and H+, allowing the transformation τ H+ to modify P effectively Conversely, the transformation τ H− is symmetrically defined, and while τ H+ cancels the effects of τ H− when applicable, they are not mutual inverses Instead, we refer to τ H+ and τ H− as mutual reverses By considering the set of all such transformations T, we can illustrate another example of a medium represented by the pair (P, T) This concept can also be exemplified by a configuration involving five straight lines.

Figure 1.3 illustrates R², which defines fifteen states and ten pairs of mutually reverse transformations Ovchinnikov (2006) provides the proof that any arbitrary locally finite hyperplane arrangement establishes a medium, as detailed in Theorem 9.1.8 in Chapter 9.

In a two-dimensional space, five straight lines can delineate fifteen distinct states, creating ten pairs of transformation paths Notably, two direct routes from state S to state T intersect the same lines, albeit in varying sequences: lihjk and ikjhl.

The Set of Linear Orders

In this example, the 24 linear orders of the set {1,2,3,4}, represented as 4!, are considered as distinct states A transformation involves swapping two adjacent numbers, denoted as τ ij, which replaces the adjacent pair j i with the pair i j, or remains unchanged if j i is not adjacent in the initial state Therefore, there are 6 possible transformations for this configuration.

2 pairs of transformationsτ ij , τ ji Three of these transformations are ‘effective’ for the state 3142, namely:

As in the preceding example, no natural orientation arises here.

1.3.1 The Permutohedron.The graph of the medium of linear orders on

Figure 1.4 illustrates a graph known as a permutohedron, as referenced in works by Bowman (1972), Gaiha and Gupta (1977), and Le Conte de Poly-Barbut (1990) In this context, we will consistently omit loops The edges of this polyhedron are organized into six families of parallel edges, with each family representing the same pair of transformations.

Figure 1.4.Permutohedron of{1,2,3,4} Graph of the medium of the set of linear orders on{1,2,3,4}.

The family of all linear orders on a finite set is intrinsically linked to the group of permutations associated with that family As demonstrated in Figure 1.4, where parallel edges symbolize pairs of mutually reverse transpositions of adjacent objects, the notion of a medium emerges as a significant algebraic structure related to linear orders.

The Set of Partial Orders

In any finite set S, the family P of all strict partial orders exhibits a unique characteristic: any given partial order P can be transformed into another partial order P through a series of steps that involve either adding or removing ordered pairs of elements from S This transformation maintains the integrity of the family P, ensuring that the process remains within the realm of strict partial orders Notably, these adjustments can be executed in the minimal number of steps, which corresponds to the 'symmetric difference' between the two partial orders By conceptualizing each partial order as a distinct state, the transformations can be likened to the movements in a jigsaw puzzle, providing a clear orientation to the process.

Figure 1.5 illustrates the graph representing all partial orders on the set {a, b, c}, highlighting the edges related to the transformation P → P + {ba} In this context, the '+' symbol signifies a disjoint union A significant characteristic of certain oriented media is their 'closed' nature concerning orientation This closure property is evident in the graph: if P, P + {xy}, and P + {zw} are defined as three partial orders on the set {a, b, c}, then P + {xy} + {zw} also qualifies as a partial order.

This medium exemplifies the parallelism property seen in the per- mutohedron, where each set of parallel edges corresponds to a pair of mutually reverse transformations between adjacent objects This characteristic of specific media is further examined in Definition 2.6.4 and Theorem 2.6.5.

Chapter 5 explores families of relations, including 'biorders' and 'semiorders,' through the lens of media theory, with specific emphasis on partial orders as illustrated in Definition 1.8.3 and Theorem 5.3.5.

8 1 Examples and Preliminaries a b c b a c a c b c a b c b a b c a a c b c a b c a b b c a a a c a b b c a c b c b a b c b c a b c b a a b c c a ba ba ba ba ba

Figure 1.5 illustrates the graph representing the medium of all partial orders on the set {a, b, c} In this graph, the orientation of the edges signifies the addition of a pair to a partial order, with one class of edges specifically labeled to indicate the addition of the pair (b, a) For further details on this notation, refer to section 1.8.2.

An Isometric Subgraph of Z n

An effective geometric representation of a finite oriented medium is as an isometric subgraph of the n-dimensional integer lattice \( Z^n \), where \( n \) is minimal The term 'isometric' indicates that the distance between vertices in the subgraph matches that in \( Z^n \) This type of representation is always feasible, as supported by Theorems 3.3.4, 7.1.4, and 8.2.2, and there are algorithms available for its construction, detailed in Chapter 10.

In a finite oriented medium M represented by a subgraph G ⊂ Z n, each state corresponds to a vertex in G Mutually reverse transformations (τ, τ) are linked to a hyperplane H that is orthogonal to one of the coordinate axes, denoted as q j The intersection of H with q j occurs at the point (i 1 , , i j , , i n) Identifying M with G establishes M = G The transformation τ, when restricted to the intersection H∩G, acts as a one-to-one function that shifts H∩G one unit upward, while remaining the identity function on the complement G\H Conversely, the reverse transformation ˜τ moves H∩G one unit downward and maintains the identity on G\H.

Figure 1.6.An isometric subgraphDofZ 3 The orientation of the induced medium corresponds to the natural order of the integers and is indicated by the three arrows.

To avoid cluttering the graph, the labeling of some edges is omitted.

The oriented graph depicted in Figure 1.6 illustrates a medium consisting of 23 states and 8 pairs of mutually reverse transformations This graph serves as a specific example of the discussed scenario, with arrows labeled 1, 2, and 3 denoting the orientations of the axes q1, q2, and q3 of Z3 Additionally, the plane defined by the vertices A, B, and D further emphasizes the structural relationships within the graph.

The transformation τ c shifts the intersection of the sets and D one unit to the right and upward, while its reverse transformation, ˜τ c, undoes this movement The eight edges marked c correspond to the pair of transformations (τ c, ˜τ c), with τ c visually represented by loops in other contexts.

The medium denoted by D is open, as the transformations τ f and τ c convert D into F and B, respectively However, applying τ c to F and τ f to B results in loops It is important to note that the subgraph D is not the sole representation of the medium M in Z 3.

In this chapter, we explore how the orientation of graphs affects the representation of edges, specifically noting that while the four edges marked as parallel to the third coordinate axis can be adjusted without compromising accuracy, a similar approach cannot be applied to the two edges marked differently The final examples provided focus on empirical applications.

In both of these cases, the medium is equipped with a natural orientation.

Learning Spaces

Doignon and Falmagne (1999) define a knowledge structure as a collection K of subsets derived from a fundamental set Q of knowledge items Each subset within K represents a knowledge state, reflecting the competence level of an individual within a specific population It is established that both the empty set and the complete set Q are included in K Two essential learning axioms underpin this concept.

[K1] IfK ⊂L are two states, with |L\K| =n, then there is a chain of states

In a learning framework, the states are represented as a sequence where K0 is a subset of K1, which in turn is a subset of K2, and so on, up to Kn equating to L Each state Ki is formed by adding a new item qi from a set Q to the previous state Ki−1, indicating a disjoint union This implies that if the learner's current state K is part of a larger state L, the learner can progress to state L by acquiring knowledge one item at a time.

[K2] IfK⊂Lare two states, withK∪{q} ∈Kandq /∈L, thenL∪{q} ∈K.

In words:If itemqis learnable from stateK, then it is also learnable from any stateLthat can be reached fromKby learning more items.

A learning space, defined by Axioms [K1] and [K2], is a knowledge structure that acts as a medium for knowledge states This medium allows for transformations by adding or removing items from a state, represented by the functions τ_q and ˜τ_q This framework leads to the concept of a ‘closed rooted medium,’ enhancing our comprehension of learning spaces In combinatorics, a learning space is referred to as an 'antimatroid,' a structure first introduced by Dilworth in 1940 An empirical application of these concepts in educational settings is discussed in Section 13.1.

3 In particular, learning spaces provide the theoretical foundation for a widely used internet based system for the assessment of mathematical knowledge.

4 In a scholarly context, an ‘item’ might be a type of problem to be solved, such as

A Genetic Mutations Scheme

This chapter concludes with an example involving artificial states represented by linear gene arrangements on a segment of chromosome 5 We examine four pairs of transformations that correspond to mutations leading to chromosomal aberrations, as observed in Drosophila melanogaster (Villee, 1967) The normal gene sequence is defined as A-B-C.

B and C are three genetic segments The four mutations are listed below.

Table 1.1.Normal state and four types of mutations

B-A-C inversion of segmentAB a From another chromosome.

1.7.1 Mutation Rules.These mutations occur in succession, starting from the normal state A-B-C, according to the five following (fictitious) rules: [IN] The segment A-B can be inverted whenever C is not duplicated.

[TR] The translocation of the segment X can only occur in the case of a two segment (abnormal) state.

[DE] A single segment C can always be deleted (from any state).

[DU] The segment C can be duplicated (only) in the normal state.

[RE] All the reverses of these four mutations exist, but no other mutations are permitted.

Figure 1.7 illustrates the graph of a medium, considering the potential for reverse mutations in all instances If these reverse mutations are infrequent, it can be inferred, within the context of the random walk process outlined in Chapter 12, that some or all reverse mutations may occur with a very low positive probability.

5 This example is inspired by biogenetic theory but cannot be claimed to be fully faithful to it Our goal here is only to suggest potential applications.

12 1 Examples and Preliminaries du i de i de i t t

The oriented graph illustrated in Figure 1.7 represents the medium affected by the four mutations identified in Table 1.1, following the five rules outlined in section 1.7.1 In this graph, the edge labels indicate specific mutations: 'i' denotes the inversion of A-B, 'du' signifies the duplication of C, 'de' represents the deletion of C, and 't' is used for additional transformations.

Notation and Conventions

We briefly review the primary mathematical notations and conventions em- ployed throughout this book A glossary of notation is given on page 309.

In set theory, standard logical and set theoretical notation is employed consistently Logical equivalence is represented by ⇔, while implication is denoted by ⇒ The symbol ⊆ indicates the inclusion of sets, whereas ⊂ signifies proper or strict inclusion Additionally, the union of disjoint sets may be expressed using the plus sign (+) or the summation symbol.

The union of all the sets in a familyFof subsets is symbolized by

∪F={x x∈Y for someY ∈F}, (1.3) and the intersection of all those sets by

Defined terms and statement of results are set inslanted font Thecomple- mentof a set Y with respect to some fixed ground setXincluding Y is the setY =X\Y.

The power set of a set Z, represented as P(Z), includes all possible subsets of Z From the definitions, we observe that the union of P(Z) equals Z, while the intersection of P(Z) results in the empty set, as the empty set is an element of P(Z) Additionally, we denote the collection of all finite subsets of Z as Pf(Z).

The cardinality of a set, denoted as |X|, refers to the size of that set When two sets have the same cardinal number, they are considered equipollent Additionally, the symmetric difference between two sets X and Y is defined as the set containing elements that are in either set but not in their intersection.

Thesymmetric difference distance of two setsX andY is defined by d(X, Y) =|XY| (1.5)

If Z is finite and the function d is defined for all X, Y in P(Z), then the pair (P(Z), d) constitutes a metric space A metric space is defined as a combination of a set X and a real-valued function d on X × X, which must satisfy three key conditions for all elements x, y, and z in X.

[D1]d(x, y)≥0, withd(x, y) = 0 if and only ifx=y(positive definiteness); [D2]d(x, y) =d(y, x) (symmetry);

TheCartesian product of two setsX andY is defined as

X×Y ={(x, y) x∈X&y∈Y} where (x, y) denotes an ordered pair and & means the logical connective

‘and.’ Writing⇔for ‘if and only if,’ we thus have

More generally, (x 1 , , x n ) denotes the ordered n-tuple of the elements x 1 , , x n , and we have

The symbolsN,Z, Q, and Rstand for the sets of natural numbers, integers, rational numbers, and real numbers, respectively;N 0 andR + denote the sets of nonnegative integers and nonnegative real numbers respectively.

A binary relation R is defined as a set of ordered pairs (x, y) that belongs to the Cartesian product of two sets X and Y, represented as R ⊆ X × Y In this context, the notation xRy is commonly used to indicate that the pair (x, y) is an element of the binary relation R.

The qualifier ‘binary’ is often omitted IfR⊆X×X, thenR is said to be a binary relationonX The (relative)product of two relationsR andS is the relation

(in which ∃denotes the existential quantifier) IfR=S, we writeR 2 =RR, and in general R n +1 =R n R forn∈ N By convention, if R is a relation on

X, thenR 0 denotes theidentity relationonX: xR 0 y ⇐⇒ x=y.

Note that whenxRyandyRz, we sometimes writexRyRzfor short Elemen- tary properties of relative products are taken for granted For example: ifR,

1.8.3 Order Relations.A relation is aquasi orderon a setXif it isreflexive andtransitiveonX, that is, for all x,y, andz in X, xRx (reflexivity) xRy&yRz =⇒ xRz (transitivity)

(where ‘⇒’ means ‘implies’ or ‘only if’) A quasi orderRis apartial order on

X if it isantisymmetric onX, that is, for all xandy inX xRy &yRx =⇒ x=y.

A partially ordered set is defined as a pair (X, R) where R is a partial order on the set X In this context, a strict partial order is characterized by its transitive and asymmetric properties, meaning that for any elements x and y in X, if x is related to y (xRy), then y cannot be related to x (¬(yRx)) Additionally, the Hasse diagram represents the covering relation of a partial order (X, R), denoted as ˘R, which captures the relationships among elements in the set.

The relation \( x, zRx \) together with \( zRy \) implies that either \( x = y \) or \( y = z \), indicating that \( x \) covers \( z \) In cases where \( X \) is infinite, the Hasse diagram may be empty Conversely, the relation \( \sim R \) offers a concise and accurate representation of \( R \).

R=∪ ∞ n =0 R˘ n = ˘R 0 ∪R˘∪ ã ã ã ∪R˘ n ∪ ã ã ã (1.8) The r.h.s (right hand side) of (1.8) is called thetransitive closure of ˘R The

A Hasse diagram of a relation R represents the minimal relation whose transitive closure restores R Typically, the transitive closure of a relation Q is denoted as Q∗ Consequently, this allows us to reformulate the initial equality in the context of transitive relations.

(1.8) in the compact form R = ˘R ∗ The Hasse diagram of a strict partial order can also be defined (see Problem 1.8).

A partial orderLonX is alinear order if it is isstrongly connected, that is, for allx,y inX, xLy or yLx (1.9)

A relationLonX is astrict linear order if it is a strict partial order which is connected, that is, (1.9) holds for all distinct x,y in X.

Suppose that L is a strict linear order on X A L-minimal element of

Y ⊆X is a pointx∈Y such thatơ(yLx) for anyy∈Y A strict linear order

L on X is a well-ordering of X if every nonempty Y ⊆X has a L-minimal element In such case, we may say thatLwell-orders X.

We follow Roberts (1979) and call astrict weak orderon a setX a relation

≺onX satisfying the condition: for allx, y, z∈X, x≺y ⇒ ơ(y≺x) and either x≺z orz≺y(or both) (1.10) For the definition of a weak order, see Problem 1.10.

1.8.4 Equivalence Relations, Partitions.A binary relationRis anequiv- alence relation on a set X if it is reflexive, transitive, and symmetricon X, that is, for allx,y inX, we have xRy ⇐⇒ yRx.

The following construction is standard LetRbe a quasi order on a setX.

Define the relation∼onX by the equivalence x∼y ⇐⇒ (xRy &yRx) (1.11)

It is easily seen that the relation∼is reflexive, transitive, and symmetric on

In the context of set theory, an equivalence relation ∼ on a set X allows us to define subsets known as equivalence classes For any element x in X, the set x consists of all elements y in X such that x is related to y under the relation ∼ The collection of these equivalence classes forms the partition of X, denoted as Z = X/∼ This partition satisfies three essential properties that characterize its structure.

A family of subsets of a set X that meets the criteria [P1], [P2], and [P3] constitutes a partition of X, with the subsets referred to as the classes of the partition When a partition consists of only two classes, it is commonly known as a bipartition.

In our puzzle Example 1.1.1, we illustrate a bipartition with the family {T +, T −}, where T + encompasses all transformations that involve adding pieces to the puzzle, while T − includes the corresponding transformations that remove those pieces.

Graphs are an integral part of graph theory, which closely aligns with the language of relations This connection becomes particularly evident when employing geometric representations We will utilize either graphs or relations as the context requires.

Adirected graphordigraphis a pair (V, A) whereV is a set andA⊆V×V.The elements of V are referred to as vertices and the ordered pairs in A as

A directed graph, or digraph, is defined by a pair (V, A), where V represents the set of vertices and A represents the directed edges In visual representations, such as in Figure 1.6, the vertices can be depicted as all possible partial orders on the set {a, b, c}, with arcs illustrating the addition of ordered pairs to these partial orders Arcs are typically shown as arrows, while vertices are represented by points or small circles When an arc vw exists, it indicates a directed connection from vertex v to vertex w Two distinct vertices, v and w, are considered adjacent if either arc vw or arc wv is present Additionally, loops—arrows that connect a vertex to itself—are often omitted in graphical representations to avoid redundancy.

This monograph presents numerous examples of digraphs that meet a specific condition: for every arc from vertex v to vertex w, there exists a corresponding arc from w to v, indicating a symmetric relationship between the arcs.

In graph theory, a digraph is known as an agraph when it consists of pairs of arcs, denoted as (vw, wv), which are referred to as edges represented by {v, w} The permutohedron illustrated in Figure 1.4 serves as a geometric representation of such a graph Typically, a single line is used to connect two points that represent adjacent vertices, simplifying the depiction of the two arcs.

Let s n be a sequencev 0 , v 1 , , v n of vertices in a digraph such thatv i v i +1 is an arc, for 0≤i≤n−1 Such a sequence is called awalk from v 0 tov n

A segment of the walk s n is a subsequences j , s j +1 , , s j + k , with 0≤ j ≤ j+k ≤ n The walk s n is closed if v 0 = v n , and open otherwise A walk whose vertices are all distinct, is apath A closed path is acircuitor acycle.

Historical Note and References

The concept of a medium, introduced by Falmagne in 1997, generalizes the conditions met by specific families of relations, including all strict partial orders on a finite set A crucial characteristic of these families is well-gradedness, which indicates that for any two partial orders, there exists a necessary sequence connecting them.

≺ 1=≺,≺ 1 , ,≺ n =≺ (1.12) of partial orders in Psuch that any two consecutive partial orders in (1.12) differ by exactly one pair,

|≺ i ≺ i +1 |= 1 (i= 1, , n−1), (1.13) and, moreover, such a path between ≺and ≺ in P is minimal in the sense that:

Chapter 5 highlights a property that is also met by various relation families, including semiorders and biorders, as defined in Definition 5.1.1 and illustrated in Formulas (5.13) and (5.9) This relationship was demonstrated by Doignon and Falmagne.

In 1997, the concept of 'well-gradedness' in certain families was introduced, which intriguingly relates to the transitivity of specific semigroups This connection emerged through the development of random walk models applicable in social sciences The computation of asymptotic probabilities for the states of these random walks revealed the necessity of proving a similar limit theorem Falmagne's 1997 work on the axiomatization of the medium concept laid foundational results, which were later expanded upon by Falmagne and Ovchinnikov in 2002, as well as by Ovchinnikov.

(2006) (see also Ovchinnikov and Dukhovny, 2000).

Eppstein and Falmagne (2002) conducted an algorithmic investigation of media, revealing a tight bound for the shortest 'reset sequence' and presenting a near-linear time algorithm to test for closed orientations, along with a polynomial time algorithm for finding such orientations These findings are detailed in Chapter 10 Earlier, graph theorists studied concepts similar to media, known as 'partial cubes,' which are isometric subgraphs of hypercubes, with significant contributions from Graham and Pollak (1971), Djoković (1973), and Winkler (1984) The relationship between partial cubes and media is explored in Chapter 7, highlighting critical differences in language and notation, as media theory aligns more closely with automata theory The choice of language and notation can significantly influence the outcomes of research, as illustrated throughout this volume.

Random walk models have been utilized to analyze opinion polls, particularly those related to the US presidential election These polls are conducted multiple times on a large sample of respondents, generating what is known as 'panel data.' Respondents are consistently asked to rank the candidates, resulting in a finite sequence of responses across k polls taken at various times t1, t2, , tk.

The data represented by ≺ t 1 , ,≺ t k indicates a specific type of order relation, such as weak orders This structure can be interpreted as reflecting the visits made by each respondent to various states within a random walk across the entire spectrum of these order relations.

In this framework, real-time visits are captured as a sequence of 'snapshots' that reflect a respondent's opinions at specific times Various random walk models have been employed to analyze the outcomes of the 1992 US presidential election, which featured candidates Clinton, Bush, and Perot (Falmagne et al., 1997; Regenwetter et al., 1999; Hsu and).

The theory of random walks on various media is comprehensively discussed in the works of Falmagne (1997) and Falmagne et al (2007), with detailed insights presented in the final two chapters of our article.

Media theory plays a crucial role in understanding learning spaces, defined as 'rooted, closed mediums.' These learning spaces serve as a framework for organizing feasible knowledge states within subjects like algebra or chemistry The specific learning space related to a topic is fundamental to developing algorithms that assess knowledge in that area.

The system effectively guides and monitors student learning, and is widely implemented in schools and colleges both in the US and internationally For more in-depth information on this topic, please refer to Chapter 13.

In view of the wide diversity of the examples covered in this chapter and later in this book, it seems likely that media have other useful applications.

This book will thoroughly address many of the questions posed below, with certain issues requiring the application of specific axioms related to a medium, as outlined in Definition 2.2.1.

In such cases the reader should rely on the intuitive conception of a medium developed in this chapter to analyze the problem and attempt a formalization.

We propose such exercises as a useful preparation for the rest of this volume.

In the Gauss puzzle 1.1.1, identify two states, S and T, along with two distinct minimally short paths connecting them These paths should not have their transformations executed in exact opposite orders To ensure that the transformations occur in exact opposite orders, specific necessary and sufficient conditions must be met.

1.2 A facet of the permutohedron on four elements is either a square or a regular hexagon Describe the facets of a permutohedron on five elements.

Certain oriented media, like the puzzle example, are considered 'closed' due to their specific orientation properties In such media, when any state S undergoes two positive transformations, τ and à, resulting in distinct states Sτ and Sà, the combination of these transformations (Sτ)à and (Sà)τ yields a state that remains within the medium This characteristic defines a closed medium To determine if a medium induced by a finite set of straight lines in the plane is closed, one must analyze the conditions outlined in Section 1.2.1 and Figure 1.3.

6 The exact connection between learning spaces and media was recognized only recently.

1.4 LetR be a strict partial order Suppose that H is a family of relations such that, for each Q ∈ H, the transitive closure Q ∗ of Q is equal to R.

Verify that we have then∩H= ˘R (So, the phrase ‘the smallest relation the transitive closure of which gives backR’ makes sense in 1.8.3.)

1.6 Prove the inclusion (1.7) Why don’t we have the equality?

The medium depicted in Figure 1.7 features a specific orientation and is not closed under this orientation Additionally, it lacks the state A-B-C-X To create a closed medium that includes all eight states from Figure 1.7, along with the state A-B-C-X and potentially two additional states, modifications to the existing rules are necessary.

1.8 Define the Hasse diagram of a strict partial order Exactly when is the Hasse diagram of a partial order or strict partial order empty?

The closure property of the medium for partial orders on the set {a, b, c} indicates that if P, P + {xy}, and P + {zw} are all partial orders within the same medium, then the combination P + {xy} + {zw} also forms a partial order This prompts the investigation of whether this closure property is applicable to all families of partial orders on any finite set, necessitating either a proof of its validity or a counterexample to demonstrate its limitations.

1.10 In the spirit of the distinction between a partial order and a strict partial order (Definition 1.8.3), define the concept of a weak orderon a setX.

Token Systems

2.1.1 Definition.Let S be a set of states A token (of information) is a function τ : S → Sτ mapping S into itself We write Sτ = τ(S), and

Sτ 1 τ 2 ã ã ãτ n = τ n (ã ã ãτ 2 (τ 1 (S))ã ã ã) for the function composition By defini- tion, the identity function τ 0 on S is not a token Let T be a set of tokens onS, with|S| ≥2 andT =∅ The pair (S,T) is then called atoken system.

LetV andS be two states ThenV isadjacenttoS ifS=V andSτ=V for some token τ in T A token ˜τ is a reverse of a token τ if for any two adjacent statesS andV, we have

A token can have at most one reverse, and if a reverse exists, it is mutually exclusive with the original token Consequently, the relationship between the token and its reverse is symmetric, meaning that the reverse of the reverse returns to the original token.

2.1.2 Definition.A message is a (possibly empty) string of elements of the set of tokens T A nonempty message m = τ 1 τ n defines a function

In the context of state transitions, we denote a message \( m \) as \( m = \tau_1 \ldots \tau_n \) within a set of states \( S \) The notation \( Sm = V \) indicates that the message \( m \) produces \( V \) from state \( S \) The content of the message \( m \), represented as \( C(m) = \{\tau_1, \ldots, \tau_n\} \), consists of its tokens, while the length of the message is denoted as \( |m| = n \), ensuring that \( |C(m)| \leq |m| \) A message is considered effective for a state \( S \) if \( Sm = S \), and ineffectiveness is indicated by \( Sm = S \) Additionally, a message is stepwise effective for \( S \) if it satisfies \( S\tau_1 \ldots \tau_k = S\tau_0 \ldots \tau_{k-1} \) for \( 1 \leq k \leq n \) Messages that are both stepwise effective and ineffective for a state are classified as return messages.

A message is deemed inconsistent if it includes both a token and its reverse, indicated by the presence of distinct indices i and j where τ j equals ˜τ i Conversely, if a message does not contain such pairs, it is considered consistent Notably, a message composed of a single token is always consistent.

Two messages, m and n, are considered jointly consistent if the combination of m and n is consistent A message that is consistent and stepwise effective for a specific state S, while ensuring that none of its tokens appear more than once, is termed concise.

(forS) A message m =τ 1 τ n isvacuousif the set of indices{1, , n}can be partitioned into pairs{i, j},suchτ i andτ j are mutual reverses.

When a message m is empty, it is defined as m = τ 0, representing the identity on S, which is not considered a token In this context, m serves as a placeholder symbol that can be removed For example, we can express this as: "let mn be a message where m is either a concise message or empty," leading to the conclusion that mn = n.

Axioms for a Medium

2.2.1 Definition.A token system (S,T) is called a medium (on S) if the following two axioms are satisfied.

[Ma] For any two distinct statesS, V inS, there is a concise message pro- ducing V fromS.

[Mb] Any return message is vacuous.

A medium (S,T) is finite if S is a finite set For some comments on these axioms, see Remark 2.2.8.

2.2.2 Lemma In a medium, each token has a unique reverse.

For any token τ, it can be established that Sτ equals V for distinct states S and V According to Axiom [Ma], there exists a concise message m that transforms V into S Consequently, τm serves as a return for S Axiom [Mb] indicates that τm must be vacuous, leading to the conclusion that ˜τ belongs to C(m) Since m is defined as concise, it follows that m is equivalent to ˜τ, demonstrating uniqueness based on the definition of a reverse token.

2.2.3 Definition.The reverse of a message m = τ 1 τ n is defined by m = ˜τ n ˜τ 1 The following facts are straightforward (see Problem 2.1): (i) if m is stepwise effective for S, then Sm = V implies Vm = S;

(ii)τ∈C( m ) if and only if ˜τ ∈C( m ); (iii) if m is consistent, so is m

Figure 2.1.Digraph of a medium with set of statesS={S, V, W, X, T}and set of tokensT={τ i 1≤i≤6}.

Figure 2.2 Digraphs of two token systems establishing the independence of the two Axioms [Ma] and [Mb] Each digraph is labelled by the failing axiom.

2.2.4 Example.Figure 2.1 displays the digraph of a medium with set of statesS ={S, V, W, X, T} and set of tokens T ={τ i 1 ≤i ≤6} It is clear that ˜τ 1 =τ 2 , ˜τ 3 =τ 4 , and ˜τ 5 =τ 6

2.2.5 Theorem The axioms[Ma]and[Mb]are independent.

The two digraphs illustrated in Figure 2.2 on page 25 each establish a token system that adheres to one of the two axioms that define a medium The labels [Ma] and [Mb] associated with the digraphs signify the specific axiom that is not satisfied (refer to Problem 2.3).

2.2.6 Convention.Except when stated otherwise, we assume implicitly for the rest of this chapter that we have fixed a token system in which Axioms [Ma] and [Mb] hold.

In the context of consistent messages, if two messages, n and m, produce the same state S and are stepwise effective for potentially different states T and V, then these messages are considered jointly consistent.

If T equals V, then the message nm serves as a return for T and must be vacuous, as stated in [Mb] Assuming nm is inconsistent, there exists a token τ that appears in both n and m Given that nm is vacuous, the token ˜τ must appear multiple times, which contradicts the consistency of n and m.

If T equals V, Axiom [Ma] indicates that a concise message w can produce T from V, leading to the conclusion that n mw serves as a return for T However, according to [Mb], n mw is vacuous If nm lacks consistency, there exists a token τ in C(n) with its counterpart ˜τ in C(m), suggesting that τ is repeated at least twice in nm Given that n mw is vacuous and both n and m are consistent, it follows that the token ˜τ must also appear at least twice in w, which contradicts the premise that w is concise.

Earlier discussions on the concept of a medium, as outlined by Falmagne (1997) and Falmagne and Ovchinnikov (2002), relied on a less stringent interpretation of [Ma], which only required a consistent message to be produced from any two distinct states, S and V In this context, Lemmas 2.2.2 and 2.2.7 were treated as axioms, along with a variation of Axiom [Mb].

All the examples of media encountered so far were finite ones Our first infinite example is given below.

2.2.9 Example.Let (Z 2 ,T) be the medium with the set of transformations

Tcontains all pairs of tokensτ ij , ˜τ ij defined by τ ij :Z 2 →Z 2 : (k, q)→(k, q)τ ij ⎧⎪

(k+ 1, q) ifi=k, j= 1, (k, q+ 1) ifi=q, j= 2, (k, q) otherwise, ˜ τ ij :Z 2 →Z 2 : (k, q)→(k, q)˜τ ij ⎧⎪

It is easily checked that Axioms [Ma] and [Mb] hold Note that both the set of states and the set of tokens are infinite (see Problem 2.9).

Preparatory Results

Some simple consequences of the axioms are gathered in the next lemma.

2.3.1 Lemma.(i)No token can be identical to its own reverse.

(ii)Any consistent, stepwise effective message (for some state) is concise.

(iii) For any two adjacent states S and V, there is exactly one token producingV from S.

(iv)Letmbe a message that is concise for some state, then

(v)No token can be a 1-1 function.

(vi)Suppose thatmandnare stepwise effective forSandV, respectively, with Sm = V and Vn = W Then mn is stepwise effective for S, with

(vii)Any vacuous message which is stepwise effective for some state is a return message for that state.

Notice that the last statement is a partial converse of Axiom [Mb].

Proof.(i) Suppose thatτ= ˜τ Asτ =τ 0 , the identity function onS, we must have Sτ=V for some distinct statesS andV This impliesSτ˜=V, and so bothτ and ˜τ produceV, contradicting Lemma 2.2.7.

(ii) Suppose that m is a consistent, stepwise effective message producing

V from S, and that some tokenτ occurs at least twice in m We have thus

In the context of effective messaging, we find that Sm can be expressed as the product of consistent stepwise effective messages n1τ and n2τn3, equating to a state V It follows that Sm is also equal to Sn1τn2τ, indicating the existence of a state V This leads us to conclude that the messages n1τ and n2 are consistent, stepwise effective messages that yield the same state.

W =Sn 1 τ˜ from S andV, respectively These two messages are not jointly consistent, contradicting Lemma 2.2.7.

Suppose that Sτ 1 = Sτ 2 = V The message τ 1 τ˜ 2 is a return for S By

[Mb], this message is vacuous, so{τ 1 ,τ˜ 2 }is a pair of mutually reverse tokens.

(iv) Equations (2.2) and (2.3) stem readily from the definition of a concise message.

(v) Suppose thatSτ =V for some tokenτ and two adjacent statesS and

V IfV τ =W =V for some state W, then V =Sτ =Wτ, a contradiction˜ of Lemma 2.2.7, because, by definition, τ is a consistent message Hence,

Sτ =V τ =V, and soτ is not a 1-1 function.

(vii) Let m be a vacuous message which is stepwise effective for some state

In the context of the relationship between S and V, where S equals V, Axiom [Ma] indicates the existence of a concise message n that generates S from V Consequently, mn serves as a return for S, which must be vacuous according to [Mb] However, since m is vacuous, it follows that n must also be vacuous, contradicting the premise that n is a concise message.

We introduce two graph-theoretical concepts The first one has been used informally in Chapter 1.

The adjacency graph of a medium is defined by vertices that represent the various states of the medium, with edges connecting vertices that indicate adjacent states These graphs are characterized as undirected.

The graphs of Figures 1.2 and 1.4 are adjacency graphs, whereas the graph of Figure 2.1 is not.

2.3.3 Definition.For a medium (S,T), we define the function δ:S×S→N0: (S, V)→δ(S, V) ⎧⎪

It is easily verified that δ is the graph theoretical distance on the graph of (S,T) (cf 1.8.5).

While the composition of tokens is not commutative in a medium, a weaker form of it holds, expressed by the next result, which is crucial.

2.3.4 Theorem Letmandnbe two distinct concise messages transforming some stateS Then

Proof.Suppose thatSm =Sn =V Then,Vn =S, which yieldsSm n =S.

The sequence m n is ineffective for state S, and according to Lemma 2.3.1(vi), it is stepwise effective for that state, confirming that m n serves as a return for S Based on Axiom [Mb], m n must be considered vacuous For any τ in C(m), since m n is vacuous, it follows that ˜τ belongs to C(m n) and, due to the consistency of m, ˜τ is also in C(n) This leads to the conclusion that τ is in C(n), establishing that C(m) is a subset of C(n) By symmetry, it can also be inferred that C(n) is a subset of C(m).

If C(m) equals C(n) with Sm, V, and Sn all equal to W, then by Axiom [Ma], there exists a concise message p that generates W from V Consequently, mp n ˜ serves as a return for S However, Axiom [Mb] suggests that mp n ˜ is vacuous, which contradicts the fact that p is concise Therefore, we can conclude that V must equal W.

Content Families

In the context of a medium defined by the pair (S, T), the content of a state S is characterized as the collection of tokens that are included in at least one concise message that generates the state S.

S={τ ∈T ∃V ∈Sand m concise such thatVm =S andτ∈C( m )}.

We refer to the familySof all the contents of the states inS as thecontent familyof the medium (S,T).

2.4.2 Example.As an illustration, we construct the content family of the medium represented in Figure 2.1 We obtain

Note the following fact regarding this family Take any two adjacent states in the Example of Figure 2.1, sayXτ 3 =W We have

2.4.3 Theorem For any tokenτ and any state S, we have eitherτ ∈S or ˜ τ ∈S(but not both); so, |S|=|V|for any two states S andV Moreover, if

Sis finite, then |S|=|T|/2for any S∈S.

Proof Since τ is a token, there are states Q = W such that Qτ = W By Axiom [Ma], for any stateS, there are concise messages m and n such that

S =Qm = Wn Thus, τn m is a return for Q, which must be vacuous by

[Mb] Therefore, ˜τ ∈ C( n ) or ˜τ ∈ C( m ), which yields ˜τ ∈ S or τ ∈ S As a consequence of Lemma 2.2.7, we cannot have both ˜τ ∈ S and τ ∈ S; so,

|S|=|V|for any two statesS andV The last statement is obvious.

2.4.4 Theorem IfSm =V for some concise messagem(thusS=V), then

By definition, the set V encompasses all tokens derived from concise messages, leading to the conclusion that C(m) is a subset of V Additionally, since Vm equals S, it follows that C(m) is also a subset of S According to Theorem 2.4.3, S cannot include both a token and its reverse, which results in the finding that C(m) is a subset of the difference between V and S.

To prove the converse inclusion, suppose thatτ ∈V\Sfor some tokenτ.

Then τ must occur in some concise message producing V Without loss of generality, we may assume thatW τn =V for some stateW, withτn concise.

Suppose thatW =Sand let q be a concise message producingSfromW As the message m nτ˜ q is a return for S, it must be vacuous by [Mb] Thus, we must have τ∈C( m )∪C( n )∪C( q ).

We cannot have τ belonging to C(q) or C(n), as this would lead to contradictions regarding the concise nature of τn Therefore, we conclude that τ must belong to C(m) The final equation of the theorem is derived from Lemma 2.3.1(iv) and Equation (2.3).

The caseS =W is left to the reader.

Any state is defined by its content Indeed, we have:

2.4.5 Theorem For any two statesS andV, we have

Proof.Suppose thatS=V for someS=V (The other implication is trivial.) Let q be a concise message producingV fromS From Theorem 2.4.4, we get

V S = C( q ) +C( q ) = ∅, contradicting our hypothesis that S = V We conclude that (2.4) holds.

The content family of a medium satisfies a strong structural property which will be investigated in Chapter 3.

The Effective Set and the Producing Set of a State

Theorem 2.4.5 entails a characterization of each state of a medium by its content In fact, only a possibly small subset of the content is needed to specify a state, enabling a more economical coding.

2.5.1 Definition.Theeffective setand the productive set of a state S in a medium (S,T) are the two subsets of tokens respectively defined by

The significance of these two concepts lies in the fact that any state within a medium is defined by one of these two sets The subsequent theorem is conceptually linked to findings regarding well-graded families, as discussed by Doignon and Falmagne (1999) in Theorem 2.8, along with relevant definitions and theorems presented in Chapter 5 The relationship is further elaborated in Theorem 5.2.5.

Orderly and Regular Returns

2.5.2 Theorem For any two statesSandTin a medium(S,T), the following

The proof demonstrates that Condition (i) leads to all other conditions, while Conditions (ii) and (iii) imply Conditions (iv) and (v) To establish that Conditions (iv) and (v) also imply Condition (i), we utilize contraposition If S equals T, a concise message τ1 τn can generate T from S According to Theorem 2.4.4, this situation results in both ˜τ1 being part of S E and (T\T), and τ1 being part of S P and T, which contradicts Conditions (iv) and (v).

Our presentation follows Falmagne and Ovchinnikov (2007).

An orderly return for a state S is defined as the combination of two concise messages, m and n, that produce the same state V, where V equals S According to Theorem 2.4.4, it is evident that the length of an orderly return must be even.

We begin with a result of general interest for orderly returns.

2.6.2 Theorem LetS,N,QandW be four distinct states of a medium and suppose that

N τ =S, W à=Q, Sq =Nq =Q, Sw =Nw =W (2.5) for some tokens τ and à and some concise messages q, q , w and w (see Figure2.3) Then, the four following conditions are equivalent:

Moreover, any of these conditions implies thatqà˜ wτ is an orderly return forS withSqà˜=S˜τw =W The converse does not hold.

Proof.We prove (i)⇒(ii)⇔(iii)⇒(iv)⇒(i).

In the scenario where τ equals à, the token ˜τ must appear precisely once in either q or w Given that à is equivalent to ˜τ and both q and w are concise, the message τqà˜w serves as a return for S, which is vacuous according to [Ma] It can be verified that there are two mutually exclusive and exhaustive cases to consider.

Figure 2.3 Illustration of the conditions listed in (2.5).

( q ) + ( w ) = ( q ) + ( w ), (2.6) contradicting (i) Thus, we must haveτ=à.

We only prove Case [a] The other case is treated similarly Since ˜τ is in

In the context of C(q), neither τ nor ˜τ can belong to C(q) Both q and q are concise, and the expression q q τ serves as a return for S Consequently, both ˜τq and q are concise messages that generate Q from S According to Theorem 2.4.4, it follows that C(˜τq) equals C(q), indicating that ˜τq is equivalent to q.

A argument along the same lines shows that

Adding (2.7) and (2.8) and simplifying, we obtain (2.6) The proof of Case [b] is similar.

(ii)⇔(iii) Ifà=τ, it readily follows (since both q and w are concise and

Sqτ˜ wτ =S) that any token in q must have a reverse in w and vice versa. This impliesC( q ) =C( w ), which in turn imply ( q ) = ( w ), and so (iii) holds.

As qà˜ wτ is vacuous, it is clear that (iii) implies (ii).

(iii)⇒ (iv) Since (iii) implies (ii), we haveτ ∈Q\N by Theorem 2.4.4. But both q and q are concise, soτ ∈C( q )\C( q ) Asτq q is vacuous forN, we must haveC( q ) +{τ}=C( q ), yielding

To demonstrate that à equals ˜τ, we note that (iv) is a specific instance of the first statement in (i) Assuming à = ˜τ, we must consistently assign the token ˜τ to maintain the vacuity of the messages q, q, τ, and τw, w According to Theorem 2.4.4, C(q) is equal to Q\S Given that ˜τ is an element of Q and, based on Theorem 2.4.3, ˜τ is not an element of S, this relationship holds true.

In the context of orderly and regular returns, we find that the only possibility is that τ belongs to C(q) but not C(q), and similarly for τ in C(w) but not C(w) This leads to the concise messages ˜τq and q generating Q from S, and w and τw generating W from N Consequently, we establish that (q) equals (˜τq) and (w) equals (τw) Thus, we derive that (q) is equal to (q) plus 1 and (w) is equal to (w) plus 1, resulting in (q) plus (w) equaling (q) plus (w) plus 2, which contradicts condition (iv) Therefore, condition (iv) implies condition (i), leading us to conclude that the four conditions (i)-(iv) are equivalent.

Under the theorem's hypotheses, we demonstrate that (ii) implies that qà˜ wτ serves as an orderly return for S, with Sqà˜ equating to S˜τw, which is W Both q and w are defined as concise by hypothesis It is not possible for à to exist in C(q), as this would result in ˜à being present.

The messages C(q) and the concise messages q and τ = à are not jointly consistent, leading to a contradiction as outlined in Lemma 2.2.7 Additionally, the presence of ˜ à in C(q) is not possible because the concise messages q and à producing Q would also lack joint consistency Consequently, qà˜ serves as a concise message producing W from S Similarly, with τ = à, ˜τw is identified as a concise message producing W from S Thus, we conclude that with τ = à, the message qà˜ wτ represents an orderly return for S Figure 2.4 illustrates this with the scenario where à = τ, q = α˜τ, w = ˜àα, w = α˜τà, ˜, and q = α, ultimately demonstrating the orderly return α˜τà˜˜αàτ for S This example acts as a counterexample to the implication that if qà˜ wτ is an orderly return for S, then τ must equal à.

Figure 2.4 The medium represented above (cf Problem 2.19) shows that, under the conditions of Theorem 2.6.2, the hypothesis that qà˜ wτe is an orderly return for

S does not implyτ =à, with q =ατ˜, w = ˜àα, q =α, and w =ατ˜à˜.

2.6.3 Remark.In fact, each of the conditions (i)-(iv) in Theorem 2.6.2 im- plies that, not only qà˜ wτ, but also various other messages are orderly return (see Problem 2.8).

In section 2.6.1, the orderly return concept was defined for a specific state The subsequent definition and theorem address scenarios where a vacuous message qualifies as an orderly return for all produced states In these instances, every token in a return must have its reverse positioned precisely at the corresponding opposite location Examples of this phenomenon can be observed in the 'hexagons' within the permutohedron illustrated in Figure 1.4 on page 6.

2.6.4 Definition.Letτ 1 τ 2 n be an orderly return for some stateS For any

1 ≤ i ≤ n, the two tokens τ i and τ i + n are called opposite A return τ 1 τ 2 n from S is regular if it is orderly and, for 1 ≤ i ≤ n, the message τ i τ i +1 τ i + n −1 is concise forSτ 1 τ i −1.

2.6.5 Theorem Let m = τ 1 τ 2 n be an orderly return for some state S. Then the following three conditions are equivalent.

(i)The opposite tokens ofmare mutual reverses.

(ii)The returnm is regular.

(iii)For1≤i≤2n−1, the messageτ i τ 2 n τ i −1 is an orderly return for the stateSτ 1 τ i −1

Proof.We prove (i)⇒(ii)⇒(iii)⇒(i) In what followsS i =Sτ 0 τ 1 τ i for

(i)⇒ (ii) Since m is an orderly return, for 1≤j ≤n, there is only one occurrence of the pair{τ j ,τ˜ j }in m Since ˜τ j =τ j + n , there are no occurrences of{τ j ,˜τ j } in p =τ i ã ã ãτ i + n −1, so it is a concise message forS i −1.

(ii) ⇒ (iii) Since m is a regular return, any message p = τ i ã ã ãτ i + n −1 is concise, so any token of this message has a reverse in the message q τ i + n τ 2 n τ i −1 Since p is concise and ( q ) =n, the message q is concise.

It follows that pq is an orderly return for the stateS i −1

The message sequence τ i τ 2 n τ i −1 serves as an orderly return for state S i −1, while the messages q = τ i +1 τ i + n −1 and q = τ i τ i + n −1 are concise for states S = S i and N = S i −1, respectively, leading to the state Q = S i + n −1 Similarly, the messages w = ˜τ i −1 τ˜ 2 n τ˜ i + n and w = ˜τ i ˜τ 2 n ˜τ i + n are concise for states N = S i −1 and S = S i, respectively, resulting in the state W = S i + n It follows that ( q ) + ( w ) + 2 = ( q ) + ( w ), confirming through Theorem 2.6.2 that τ i + n = ˜τ i.

2.6.6 Remark.Examples of media with returns that are not regular and yet satisfy Conditions (i)-(iv) of Theorem 2.6.2 are easily constructed (see Problem 2.13).

Following Ovchinnikov (2006), we now introduce a couple of general tools for the study of media.

Embeddings, Isomorphisms and Submedia

2.7.1 Definition.Let (S,T) and (S ,T ) be token systems A pair (α, β) of 1-1 functionsα:S→S ,β:T→T such that

Sτ=T ⇐⇒ α(S)β(τ) =α(T) (S, T ∈S, τ ∈T) is called anembeddingof the token system (S,T) into the token system (S ,T ).

If the functionsαandβ are bijections, then these two token systems are said

2.7 Embeddings, Isomorphisms and Submedia 35 to be isomorphicand the embedding (α, β) is referred to as anisomorphism of (S,T)onto(S ,T ).

In scenarios where one token system serves as a medium, the other token system must also function as a medium This is supported by Problem 2.10, which highlights that if (S,T) is classified as a medium and both τ 1 and τ 2 are elements of T, resulting in Sτ 1 = Sτ 2 = S for a specific S in S, it follows that τ 1 must equal τ 2, as established by Lemma 2.3.1(iii).

Thus, if (α, β) is an isomorphism of a medium onto another medium, we haveβ(˜τ) =β(τ) Indeed, for a givenτ there are two distinct statesS andT such thatSτ˜=T Then α(S)β(˜τ) =α(T) ⇐⇒

We extend the function β of an isomorphism (α, β) to the semigroup of messages by defining β for a message m = τ₁ τₙ as β(τ₁ τₙ) = β(τ₁) β(τₙ) This clearly shows that the output of the function β applied to a concise message remains a concise message.

In a medium defined by a pair (S, T), consider a subset Q of S that contains at least two states The collection T comprises all restrictions of tokens in T to the subset Q However, the pair (Q, T) does not always qualify as a medium An illustrative example is provided by the graph in Figure 2.1, where restricting the tokens to the subset {S, T} fails to produce a medium, as highlighted in Problem 2.11.

We introduce the appropriate concepts for the discussion of such matters.

2.7.2 Definition.Let (S,T) be a token system and let Q be a nonempty subset ofS For anyτ ∈T, the function τ Q : Q→Q : S→Sτ Q Sτ ifSτ∈Q,

S otherwise is called the reduction 1 of τ to Q Note that the reduction of a token to a subset Q of states may be the identity on Q A token system (Q,T Q ) where

The set TQ = {τ Q} where τ ∈ T \ {τ 0} represents all non-identity reductions of tokens in T to Q, and is known as the reduction of (S,T) to Q The pair (Q,TQ) is identified as a token subsystem of (S,T) When both (S,T) and (Q,TQ) qualify as media, (Q,TQ) is considered a submedium of (S,T), indicating that (S,T) is inherently a submedium of itself.

1 The concept of ‘reduction’ is closely related to, but distinct from, the concept of

‘restriction’ of a function to a subset of its domain The standard interpretation of ‘restriction’ would not remove pairs (S, Sτ) in whichS∈QbutSτ∈(S\Q).

2.7.3 Remarks.(a) The above definition readily implies that if (S,T) and (Q,T Q ) are two media, with (Q,T Q ) a reduction of (S,T) toQ, thenτ Q =à Q only ifτ =à, for any τ Q andà Q inT Q

The image of a token system (S,T) under an embedding (α, β) is not typically a reduction of (S,T) to α(S) However, Theorem 2.7.4 demonstrates that this holds true specifically for media.

2.7.4 Theorem Suppose that (α, β) is an embedding of a medium (S,T) into a medium(S ,T ) Then the reduction(α(S),T α (S) )of(S ,T )toα(S)is isomorphic to(S,T).

Proof For τ ∈ T, we use the abbreviation β (τ) = β(τ) α (S) (thus, the re- duction of β(τ) to α(S)) Let Sτ = T = S Then α(S)β(τ) = α(T) for α(S) =α(T) inα(S) Henceβ mapsT intoT α (S) Let us show that (α, β ) is an isomorphism from (S,T) onto (α(T),T α (S) ).

(i)β is ontoT α (S) Suppose thatτ α (S) =τ 0 for someτ ∈T Then, there areP=Q∈Ssuch that α(P)τ α (S) =α(P)τ =α(Q).

LetQ= mP, where m is a concise message fromP We have α(Q) =α(Pm ) =α(P)β( m ) =α(P)τ implying, by Theorem 2.3.4, thatβ( m ) =τ sinceβ( m ) is a concise message. Hence, m =τ for someτ ∈T Thus,β(τ) =τ , which implies β (τ) =β(τ) α (S) =τ α (S)

(ii)β is 1-1 Suppose thatβ (τ) =β (à) Sinceβ (τ) andβ (à) are tokens onα(S) and (S ,T ) is a medium, we haveβ(τ) =β(à) by Remark 2.7.3 (a).

(iii) Finally, for anyS, T ∈Sin andτ∈T

Oriented Media

In our first chapter, three out of five examples demonstrated that the medium had an 'orientation' influenced by the characteristics of the tokens used This concept of orientation is defined as follows.

2.8.1 Definition.Anorientationof a medium (S,T) is a partition of its set of tokens into two classesT + andT − respectively calledpositiveandnegative such that for anyτ ∈T, we have τ ∈T + ⇐⇒ ˜τ∈T −

A medium (S,T) equipped with an orientation{T + ,T − }is said to beoriented by{T + ,T − }and tokens fromT + (resp.T − ) are calledpositive(resp.negative).

In an oriented medium denoted as (S,T), the orientation is typically represented by the sets {T +, T −} The positive content of a state S, referred to as S +, consists of its positive tokens derived from the intersection of S and T +, while the negative content, S −, comprises its negative tokens from the intersection of S and T − We define two distinct families based on these sets.

In the context of message analysis, S + and S − represent the collections of positive and negative contents, respectively Messages composed solely of positive tokens are classified as positive, while those with only negative tokens are termed negative Two messages are considered to have the same sign if they are both positive or both negative, a classification that also extends to individual tokens It's important to note that a finite medium (S, T) can exhibit 2 |T| / 2 distinct orientations.

The definition in Remark 2.8.2 aligns with a scenario where two oriented media, (S,T) and (Q,L), are isomorphic, yet their orientations, {T +, T −} and {L +, L −}, do not correspond For instance, an isomorphism (α, β) can exist between (S,T) and (Q,L), where an element τ belongs to T + while its image under β, β(τ), is found in L −.

In fact, if (S,T) and (Q,L) are isomorphic, they would remain so under any changes of orientations There are cases in which a more demanding concept is required.

2.8.3 Definition.Two oriented media (S,T) and (Q,L) are said to besign- isomorphic if there is an isomorphism (α, β) of (S,T) onto (Q,L) such that β(T + ) =L + (and soβ(T − ) =L − ).

Let (S,T) be a medium with an orientation {T + ,T − } Then any sub- medium (Q,T Q ) of (S,T) has an induced orientation{T Q + ,T Q − } defined from

Except when indicated otherwise, a submedium of an oriented medium is implicitly assumed to be equipped with its induced orientation.

As a consequence of Theorems 2.4.3 and 2.4.5, we have:

2.8.4 Theorem For any two statesS andV of an oriented medium(S,T), we have

A similar result holds obviously for negative contents.

Proof The necessity is trivial Regarding the sufficiency, note that if both

S=S + +S − =V + +V − =V , and so S = V by Theorem 2.4.5 It suffices thus to prove that S + = V + implies S − = V − Suppose that S + = V + and take any τ ∈ T We have successively τ∈S − ⇐⇒ τ˜∈T + \S + (by Theorem 2.4.3)

2.8.5 Theorem If S and V are two distinct states in an oriented medium, withSm =V for some positive concise messagem, then S + ⊂V +

Proof If m is a concise positive message producing V from S, then m is concise and negative, and S =Vm , with S\V = C( m ) by Theorem 2.4.4.Thus, any token inS\V is negative This impliesS + ⊂V +

The Root of an Oriented Medium

It is natural to ask how an orientation could be systematically constructed so as to reflect properties of the medium One principle for such a construction is described below.

An oriented medium is defined as having a root state R, where any concise message that transitions from R to another state S is considered positive When an oriented medium possesses a root, it is referred to as rooted Additionally, it is common to describe the orientation of a medium as rooted at state S if the medium is rooted for that orientation and S serves as the root state.

2.9.2 Theorem In an oriented mediumM, a stateRis a root if and only if

R + =∅ Thus,Mhas at most one root and may in fact have no root.

In a state R where R + is empty, if we consider a concise message m such that Rm equals S, then Sm equals R According to Theorem 2.4.4, R\S is equal to C(m) If m includes a positive token, it implies that R + remains empty, leading to the conclusion that m is negative, which paradoxically suggests that m is also positive This reasoning applies to any message m effective for R, confirming that the state R must be a root.

An Infinite Example

In the context of roots, if R is a root, Theorem 2.8.5 indicates that R + is a subset of S + for any state S that is not equal to R If R + includes a positive token τ, then τ is present in the contents of all states This leads to the conclusion that τ is ineffective for any state, including the identity function, which is not classified as a token Therefore, we deduce that R + must be empty.

For an example of a medium without a root, take the medium of Figure 2.1 with an orientation having the set of positive tokensT + ={τ 2 , τ 4 , τ 5 }.

2.9.3 Theorem For any medium(S,T)and any stateRinS, there exists an orientation makingRthe root of(S,T).

Proof Define such an orientation by setting T − = R Then, for any state

S = R and any concise message m producing S from R, we have C( m ) (S\R) ⊆T + by Theorem 2.4.4 Thus, m is positive.

The following type of oriented medium will play a role, in later develop- ments, as a component of larger media.

A rooted medium N = (S, T) is defined as an n-star when its root is connected to all other n-1 states, where |S| = n and n ≥ 3 This configuration results in n-1 pairs of tokens Therefore, we can classify N as a star if there exists an n ∈ N such that N meets the criteria of an n-star.

The next section contains an example in which all the sets of positive contents are uncountable We have, however, the following result.

Theorem 2.9.5 states that in any medium, it is possible to find an orientation that guarantees all states have finite positive contents Specifically, this applies to rooted mediums, where each state also possesses finite positive contents.

Proof.Indeed, by Theorem 2.9.3 we can arbitrarily assign any state to be the root R of the medium We have R + = ∅, and for any other state S, with

Rm =S for some concise message m , we haveS + = C( m ) = S\R, with

The example illustrates that both the set of states and the set of tokens are infinite, with the contents of all states being uncountable.

In this example, let Pf(R) represent the collection of all finite subsets of R We define a medium (S,T) where S consists of subsets SX, with SX being equal to R minus X, and T includes, for every real number x, the pair of mutually reverse tokens τx and ˜τx.

S∪ {x} ifS=S X for some X∈Pf , withx∈X,

S\ {x} ifS =S X for someX ∈Pf , withx /∈X,

Chapter 3 explores the concept of token systems as media, specifically highlighted in Theorem 3.3.4 In this context, both sets S and T, along with the contents of all states, are uncountable By defining an orientation where positive and negative tokens are specified, it becomes evident that while all positive contents remain uncountable, the negative contents are finite A similar counterexample arises if the parameters are adjusted accordingly.

R by any set X of arbitrary cardinality, with the set of states defined by

{S X X ∈ Pf(X), S X = X\X} The exact role played by the finite sets is discussed in details in Chapter 3.

In Example 2.10.1, it is demonstrated that positive contents can sometimes be uncountable By redefining the orientation with the set R of all real numbers, we can interchange the positive and negative tokens This results in a scenario that aligns with the second statement of Theorem 2.9.5, where the positive contents consist of all finite sets of real numbers.