A simple construction of complete single-peaked domains by recursive tiling

Single-peakedness was introduced by Black (J Political Econ 56:23–34, 1948) as a sufficient condition to overcome Condorcet paradox. Since then it has been attracting interest from researchers in various fields. In this paper, we propose a simple recursive procedure of constructing complete single-peaked domains of tiling type explicitly for any finite alternative sets, by combining two results published in recent years, and some observations of known results and examples by the author. The underlying basic structure of tiling type and properties of single-peaked domains provided here give a good visualization and make further developments on single-peakedness more easy.


Introduction
Single-peaked domains, or single-peaked preferences, were introduced by Black (1948) (single peaked curves) as a sufficient condition to overcome Condorcet paradox (Black 1948(Black , 1958, i.e., having a cyclic majority preferences. Given a single-peaked domain, a winner can be determined by a simple majority voting rule. Since then, single-peaked domains and related topics have been extensively researched, see the review paper (Monjardet 2009) and also a recently published paper by Puppe (2018).
As wrote in Fishburn (1997), and mentioned in Danilov et al. (2012): "...it is far from obvious how the restrictions should be selected jointly to produce a large acyclic B Ping Zhan zhan@edogawa-u.ac.jp 1 Department of Communication and Business, Edogawa University, 474, Komaki, Nagareyama, Chiba 270-0198, Japan sets". Our contribution is a neatly simple procedure of constructing complete singlepeaked domains of tiling type explicitly for any finite alternative sets. The underlying basic structure and properties of single-peaked domains provided here also give some insights and make further developments on single-peakedness more easy.
Our motivations for this research began with the random assignment problem (Fujishige et al. 2018). Single-peakedness is important to keep required properties for further generalization (Achuthankutty and Roy 2018;Bade 2019;Moulin 1980Moulin , 2017Savaglio and Vannucci 2019). Having a better understanding of the structure of single-peaked domains leads to the simple recursive construction provided in the following. Single-peakedness is also connected to various topics. We give these related discussions in Concluding remarks.
The paper consists of two main sections. Section 2 is divided into two subsections. Section 2.1 gives some basic definitions and recent published results, Sect. 2.2 treats rhombus tilings. Section 3 consists of three subsections, observations, procedure and theorems respectively. They provide (together with validation) the recursive construction of complete single-peaked domains of tiling type. The final section gives concluding remarks.

Single-peaked and Condorcet domains
Consider a finite set of alternatives X with a fixed linear order > on it. Let L(X ) denote the set of all preferences 1 ≺ on X . A domain D ⊂ L(X ) is single-peaked with respect to the linear order > on X if, for all (strict) preferences ≺∈ D and all x ∈ X , the upper contour sets U ≺ (x) := {y ∈ X | x ≺ y} ∪ {x} are connected ('convex') viewing from the order >. In other words, suppose thatx ∈ X is the peak/top of the preference, we have In the following, we denote alternative set as N = {1, 2, . . . , n} with the nature order. For i, j ∈ N , we write i j as i j if without confusion. 2 From now on, order refers to preference order if without special explanation.
It is known that the single-peaked domain in L(N ) contains at most 2 n−1 elements, or orderings (Black 1958;Abello and Johnson 1984). The (simple) majority relation associated with a profile π m is defined by i ≺ ma j π m j if (and only if) more voters (> m/2) prefer j than i. Here is an example of three voter's profile π 3 = (123, 231, 312) on alternatives {1, 2, 3}. It is easy to check 1 ≺ ma j π 3 3 ≺ ma j π 3 2 ≺ ma j π 3 1, i.e., a majority cycle, i ≺ Condorcet domains are characterized by the never conditions defined as follows. For h in {i, j, k} and r in {1, 2, 3}, the never conditions 4 h N i, j,k r means that h never has rank r . The single-peaked domain corresponds to, 'the middle never ranking the last'. Here 'the middle never ranking the best' is called pit domain. These two domains, will be discussed further in the following (sub)sections. For more details about CDs, please see the review paper (Monjardet 2009). Let The Bruhat digraph is formed by drawing an arc from σ to τ if and only if τ covers σ . 5 Figure 1 gives a digraph of Bruhat poset for n = 3. The four orders {123, 213, 231, 321} on the left path form a single-peaked domain, the four orders {123, 132, 321, 321} on the right path from a pit domain. Now we introduce a theorem related to single-peaked domains in paper (Puppe 2018) published recently. It is one of two main results which our recursive procedure is based on.
A domain D ⊂ L(N ) is called minimally rich if for every i ∈ N , there exists an order σ ∈ D such that σ has i as the top preference. Note that a maximal width of the digraph of Bruhat poset means that associated domain D contains at least one pair of completely reversed orders. For example, α = 12 . . . (n −1)n and ω = n(n −1) . . . 21, with Inv(α) = ∅ and Inv(ω) = Ω.
Furthermore, the domain of all orders that are single-peakedness is denoted by SP(N ). A domain D is semi-connected if it contains two completely reversed orders, and a directed path connecting them in the associated digraph of the Bruhat poset (D, ). The connectedness of the Bruhat digraph is defined in the natural way. Now we are ready to present:

Theorem 1 (Theorem 1, Puppe 2018) The domain SP(N ) of all single-peaked orders is a connected and minimally rich Condocet domain with maximal width. (In particular,
is a semi-connected and minimally rich Condocet domain, then D is single-peaked.

Representing CDs by rhombus tiling
The characterizations of Condorcet domains by the rhombus tiling given by Danilov et al. in paper (2012) and related papers (Danilov et al. 2007(Danilov et al. , 2010 develop and unify several related properties known earlier. The representation by the rhombus tiling also admits a simple and good visualization. The notations and contents of this subsection are mainly adopted from the paper (Danilov et al. 2012).
By the parallel property of arcs of rhombus tiles, tilings can be represented via a vector summation. Precisely, let ξ i ∈ R × R >0 (i ∈ N ) be vectors in the upper half-plane. The sum of segments [0, ξ i ] of (ξ 1 , ξ 2 , . . . , ξ n ) forms a zonogon, denoted by Z = Z n . This is the symmetric 2n-gon formed by the points n i−1 a i ξ i over all 0 ≤ a i ≤ 1 (1 ≤ i ≤ n), with the source vertex (point (0, 0)) at the bottom, and the sink vertex (point ξ 1 +ξ 2 +· · ·+ξ n ) at the top. A rhombus congruent to the sum of two segments [0, A rhombus tiling (or simply a tiling), denoted by T , is a subdivision of a zonogon into a set of rhombus tiles satisfying the following condition: if two tiles intersect, then their intersection consists of a single vertex or a single edge. Figure 2 shows two examples of rhombus tilings. We label the arcs of tilings with corresponding alternative i ∈ N (same labels of parallel arcs are omitted if they are obvious to readers).
Using the language of graph in the following, we associate to a tiling T with the planar directed graph G T = (V T , A T ) whose vertices and edges are those occurring in the tile T . Note that graph G T , or tiling T is an acyclic graph since all arcs are oriented upward. An maximal directed path is a path going from the source to the sink by the acyclicity of G T . We call such path a snake of T . The labels on the directed path of each snake form a linear order, or a permutation of alternatives of N (Elnitsky 1997;Danilov et al. 2012). The zonogon, or the summation of upward vector segments in a plane, is bounded by two special snakes, the left boundary lbd(Z ) and right boundary rbd(Z ) of Z , the former gives a liner order α, and the latter gives ω.
For a directed path of G T , or a linear order, if instead of two arcs of a tile on the path, by adding the other two arcs of the tile, we get a new path that differs one inversion in the linear order. Hence we can view a tile as a permutation of two adjacent, or consecutive alternatives. Moreover, right snakes have more inversion pairs than left ones, and the set of snakes is a Bruhat poset. And all snakes of a tiling also form a lattice. See more details in papers (Elnitsky 1997;Danilov et al. 2012).
Let us denote the set of linear orders associated with the snakes in tiling T by Σ(T ). In our examples shown above, Σ(T ) of tiling T in Fig. 2 associates with the set of four orders on the left path of Fig. 1; and Σ(T ), the four orders on the right path of Fig. 1.
The preference domain represented by a tiling is majority acyclic, i.e., a Condorcet domain. Since restricting every three alternatives leads to two tiles show in Fig. 2, they are peak-pit domains. Formally, a set D ⊂ L(N ) is called peak-pit domain if for each triple i < j < k in N satisfies never conditions: j never ranking the last, best, or both (Danilov et al. 2012).
We say that domain D is complete if it is inclusion-wise maximal, i.e., adding to D any new linear order violates the majority acyclicity.
Theorem 2 (Theorem 1, Danilov et al. 2012) The set Σ(T ) of tiling T is a peak-pit Condorcet domain and also complete.
Proof of the converse of Theorem 2 is less trivial. The never conditions of Condorcet domain defined in Section 2.1 relate to the separation theorems in algebra, via the reduced words of permutations, or the transpositions of symmetric groups, and Plücker functions (Leclerc and Zelevinsky 1998), also see papers (Danilov et al. 2007(Danilov et al. , 2010. The following result also has been shown in the paper (Danilov et al. 2012).

A recursive construction of complete single-peaked domains by tiling
In this section, we provide our main result, an explicit construction of complete singlepeaked domains of tiling type of any cardinality size, based on the results introduced in last section and some observations shown in the following.

Some observations of known results and examples
We first define an associated tiling for a preference domain. For a (complete peak-pit) domain D ⊂ L(N ), let us denote the associated tiling T such that Σ(T ) = D by T D .
Observation 1 Suppose that a domain D is single-peaked and T D is its associated tiling. Then, for alternative set N = {1, . . . , n}, the minimally rich condition given in Theorem 1 means that there should be n arcs connected to the source of T D . On the other hand, by the never condition of single-peaked domains, 'middle can not be the least preference', only the first and the last, {1, n}, of alternative set N can be the last element of any sink-peaked liner order. Hence, the arcs connected to the sink of T D are two alternatives only, {n} and {1}. We summarize and illustrate these facts in Fig. 3.
The above conclusions, i.e., the results have been known and obtained, do not define what are the exact tilings for single-peaked domains, and how to build these tilings. We will approach the problem with tilings of lower cardinality size of single-peaked domains.
We have seen an associated tiling T D for single-peaked domain D with n = 3, i.e., T in Fig. 2. The tiling for Example 1 given in Sect. 2.1 is drawn in Fig. 4. We also add two trivial cases, n = 1 and n = 2 for the completeness of the discussions in the following. For the single-peaked domain of size 5, it is the tiling 6 introduced by Elnitsky (1997) (Fig. 8 of the paper) as a tiling example, we redraw it in Fig. 5 here for comparing and better understanding. 7

Recursive procedure
Our construction needs notations specifically indexed by recursive procedure steps n ≥ 1.
Denote a single-peaked domain (of all orders) of tiling type with respect to alternative set N = {1, . . . , n} by SP n ⊂ L(N ). Note that we have Σ(T SP n ) = SP n . We use T R P n (n ≥ 1) for the tilings obtained from Recursive Procedure and in the proof of Theorem 4. Step 1 T R P 1 , or T SP 1 has two vertices, the source and the sink, and one vertical arc from the source and the sink (see the left diagram of Fig. 6).
To construct T R P 2 , or T SP 2 , first, add another vertical arc which parallels to, and has the same direction, length and label as T SP 1 . Then, the end of the new vertical arc becomes the new sink of tiling, and the original source remains to be the source of new tiling (see the middle diagram of Fig. 6). Draw two other parallel arcs in the same direction, both are labeled by alternative {2}, one links from the source of T SP 1 to the head of the new vertical arc, while the other links the ends of two vertical arcs (see the right diagram of Fig. 6).
Step n − 1 Identify rbd(T R P n−1 ), the right boundary of T R P n−1 constructed in the previous procedure (see the left diagram of Fig. 7). To construct T R P n , first, add another path which parallels to and has the same direction, length and labels as rbd(T R P n−1 ). Then, the end of the new path becomes the new sink, and the original source remains to be the source of new tiling (see the middle diagram of Fig. 7). Draw n other paralleling arcs in the same direction, all are labeled by alternative {n}, the first arc links from the original source to the head of the new path, while every other arc links the ends of the two paralleling arcs between rbd(T R P n−1 ) and the new path (see the right diagram of Fig. 7).
In the above Step n − 1, note that we can also change the rbd(T R P n−1 ) to lbd(T R P n−1 ), the left boundary of T R P n−1 , and add lbd(T R P n−1 ) to the left side of T R P n−1 . T R P n (n > 1) are symmetric, rbd(T R P n ), or lbd(T SP n ), is not crucial here.

Validation and properties of Recursive Procedure
In this subsection, we show that for any given alternative set N = {1, . . . , n}, T R P n obtained from Recursive Procedure satisfies Fig. 7 Step n − 1 of recursive procedure The claim is shown by two theorems, we also discuss some properties.
Theorem 4 For any alternative sets N = {1, 2, . . . , n}, 8 the tiling T SP n of singlepeaked domains SP n ⊂ L(N ) can be explicitly constructed for every integer n ≥ 1 in a recursive process.
Proof A construction procedure has been provided in Sect. 3.2. We will show that T R P n obtained from Recursive Procedure being single-peaked domains, i.e., Σ(T R P n ) = Σ(T SP n ) for any integers n ≥ 1. Since our Recursive Procedure is the CD domains of tiling type, by Theorem 2 we have Σ(T R P n ) for n ≥ 1 are peak-pit domains, We now prove that the tilings T R P n being semi-connected. It is clear that each T R P n contains two boundaries, lbd(T R P n ) and rbd(T R P n ), which are completely reversed. The connection paths between lbd(T R P n ) and rbd(T R P n ) in the digraph induced by Bruhat poset can be obtained according to a similar discussion given in Elnitsky (1997). If a snake, or a preference order, say A, is not rbd(T R P n ), there exists at least one tile in the right of snake A, having two arcs of T R P n , say i and j (i < j), common with the snake A. Replacing two arcs of i j tile in snake A with two other arcs of the tile, we obtain a new snake, say B, with Inv(B) = Inv(A) ∪ (i, j). Hence the order induced by B covers the order by A, and there is an arc from A to B in the Bruhat digraph. Beginning with lbd(T R P n ), continuing the above process until we reach rbd(T R P n ). Therefore, tiling T R P n is semi-connected.
It is easy to see that each step of Recursive Procedure increases one arc adjacent to the source. Beginning from Step 1, there are n arcs adjacent to the source of T R P n (n ≥ 1). Recall Observation 1, we know that the domain Σ(T R P n ) satisfying minimally rich condition. 9 Now all the conditions for single-peakedness needed by Theorem 1 have been satisfied.
By definition, the single-peaked domain Σ(T SP n ) is a peak-pit domain. It can be contained in a domain of tiling type as indicated in Theorem 3, which is exactly the tiling T R P n obtained from our Recursive Procedure for any integers, i.e., T R P n coinciding with T SP n for n ≥ 1.

Remark 1
The directed graph G T R Pn , or G T SPn obtained from Recursive Procedure is consist of 1 + n + (n − 1) + · · · + 1 = 1 + n(n+1) 2 vertices and n × n arcs. From the proof of Theorem 4, we know that the number of tiles in T SP n is the diameter of the Bruhat digraphs associated with single-peaked domains SP n for n > 1. The different tiles intersecting two arcs with snake A chosen in the proof of Theorem 4 leads to different sets of orders (see also Elnitsky 1997).
Here is a theorem about the completeness of SP n .
Proof We have shown in Theorem 4 that T R P n (n ≥ 1) obtained by Recursive Procedure are single-peaked domains SP n of tiling type. By Theorem 3, we know that the tiling type of CD domains is peak-pit complete. Since single-peaked domains SP n are peak-pit domains. We have SP n being complete, i.e., SP n = SP(N ) for all n ≥ 1.
Remark 2 From the proof of Theorem 5, we know that SP n (n ≥ 1) are not only complete in single-peakedness, but also complete on peak-pit domains. Let σ be a linear order, SP n ∪ {σ } is cyclic if σ is a pit order. 10 But this does not include general CD domains. Actually, for a given set of alternatives, the maximal cardinality of a CD domain is in general strictly larger than the cardinality of the single-peaked domain.
Finally, let us see how the single-peaked preferences are constructed recursively from Recursive Procedure.
Proposition 1 For each alternative set N = {1, 2, . . . , n} with n > 2, the following three types of single-peaked preferences are created recursively by T SP n in Recursive Procedure: (i) Join arc labelled by the new alternative from sink to new sink (ii) Move from source immediately to the added "rbd-copy" (iii) Move to "rbd-copy" later via "rbd".
Moreover, the number of orderings of type (i) is equal to |Σ(T SP n−1 )|.
Proof For type (i), it is clear from the construction. The operation creates all the orderings that have the new alternative at the bottom.
For types (ii) and (iii), the key point is: for each snake of T SP n , or single-peaked preference, once the snake entering "rbd-copy" newly added in Step n − 1, it ends within "rbd-copy".
Precisely, type (ii) creates the (unique) ordering that puts the new alternative on top. For type (iii), after each snake entering from a vertex of "rbd" of T SP n−1 via the arc labelled by the new alternative, it just follows the vertex of "rbd-copy" continuously. Hence, we can simply view the operation as inserting a new alternative to the orderings intersection with "rbd" in SP n−1 . Hence, type (iii) creates an ordering that puts the new alternatives somewhere in the middle.
For the part of "Moreover", since each snake, or the ordering of single-peaked of tiling type, ends at the sink.