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Journal of Homotopy and Related Structures

, Volume 10, Issue 2, pp 205–238 | Cite as

Bipermutahedron and biassociahedron

  • Martin Markl
Article
  • 83 Downloads

Abstract

We give a simple description of the face poset of a version of the biassociahedra that generalizes, in a straightforward manner, the description of the faces of the Stasheff’s associahedra via planar trees. We believe that our description will substantially simplify the notation of Saneblidze and Umble (Homology Homotopy Appl 13(1):1–57, 2011) making it, as well as the related papers, more accessible.

Keywords

Permutahedron Associahedron Tree Operad PROP Zone Diaphragm 

Mathematics Subject Classification (2000)

16W30 57T05 18C10 18G99 

1 History and pitfalls

In this introductory section, we recall the history and indicate the pitfalls of the ‘quest for the biassociahedron,’ hoping to elucidate the rôle of the present paper in this struggle.

1.1 History

Let us start by reviewing the precursor of the biassociahedron. Stasheff in his seminal paper [11] introduced \(A_\infty \)-spaces (resp. \(A_\infty \)-algebras, called also strongly homotopy or sh associative algebras) as spaces (resp. algebras) with a multiplication associative up to a coherent system of homotopies. The central object of his approach was a cellular operad \(K = \{K_m\}_{m \ge 2}\) whose \(m\)th piece \(K_m\) was a convex \((m-2)\)-dimensional polytope called the Stasheff associahedron. \(A_\infty \)-space was then defined as a topological space on which the operad \(K\) acted, while \(A_\infty \)-algebras were algebras over the operad \(C_*(K)\) of cellular chains on \(K\). Let us briefly recall the basic features of the construction of [11], emphasizing the algebraic side. More details can be found for instance in [9, II.1.6] or in the original source [11].

Consider a dg-vector space \(V\) with a homotopy associative multiplication \(\mu :{V}^{\otimes 2} \rightarrow V\). This means that there is a chain homotopy \(\mu _3 : {V}^{\otimes 3} \rightarrow V\) between \(\mu (\mu \otimes 1\!\!1)\) and \(\mu (1\!\!1\otimes \mu )\), where \(1\!\!1\) denotes the identity endomorphism \(1\!\!1: V \rightarrow V\). The homotopy \(\mu _3\) will be symbolized by the intervalconnecting the two possible products, \((ab)c\) and \(a(bc)\), of three elements \(a,b,c \in V\). We abbreviate, as usual, \((ab)c:= \mu \left( \mu (a,b),c\right) = \mu (\mu \otimes 1\!\!1)(a,b,c)\), &c. As the next step, consider all possible products of four elements and organize them into the vertices of the pentagon:The products labeling adjacent vertices are homotopic and we labelled the edges by the corresponding homotopies. Observe that all these homotopies are constructed using \(\mu _3\) and the multiplication \(\mu _2\).

Next, we require the homotopy for the associativity to be coherent, by which we mean that the pentagon \(K_4\) can be ‘filled’ with a higher homotopy \(\mu _4 : {V}^{\otimes 4} \rightarrow V\) whose differential equals the sum (with appropriate signs) of the homotopies labelling the edges. This process can be continued, giving rise to a sequence \(K = \{K_m\}_{m \ge 2}\) of the Stasheff associahedra [11], discovered independently by Tamari [13]. It turns out that \(K\) is a polyhedral operad. An \(A_\infty \)-algebra is then an algebra over the operad \(C_*(K)\) of cellular chains on \(K\).

Much later there appeared another, purely algebraic, way to introduce \(A_\infty \)-algebras. As proved in [5], the operad \(\mathcal{A }{ ss}\) for associative algebras admits a unique, up to isomorphism, minimal cofibrant model \(\mathcal{A }_\infty \) which turns out to be isomorphic to the operad \(C_*(K)\). We may thus as well say that \(A_\infty \)-algebras are algebras over the minimal model of \(\mathcal{A }{ ss}\). Finally, one can describe \(A_\infty \)-algebras explicitly, as a structure with operations \(\mu _m : {V}^{\otimes m} \rightarrow V\), \(m \ge 2\), satisfying a very explicit infinite set of axioms, see [11, page 294]. In the case of \(A_\infty \)-algebras thus topology, represented by the associahedron, preceded algebra.

There were similar attempts to find a suitable notion of \(A_\infty \)-bialgebras,1 that is, structures whose multiplication and comultiplication are compatible and (co)associative up to a system of coherent homotopies. The motivation for such a quest was, besides the restless nature of human mind, homotopy invariance and the related transfer properties which these structures should possess. For instance, given a (strict) bialgebra \(H\), each dg-vector space quasi-isomorphic to the underlying dg-vector space of \(H\) ought to have an induced \(A_\infty \)-bialgebra structure.

Here algebra by far preceded topology. The existence of a minimal model \(\mathcal{B }_\infty \) for the PROP \(B\) governing bialgebras2 was proved in [7]. According to general philosophy [6], \(A_\infty \)-bialgebras defined as algebras over \(\mathcal{B }_\infty \) are homotopy invariant concepts. Moreover, it follows from the description of \(\mathcal{B }_\infty \) given in [7] that an \(A_\infty \)-bialgebra defined in this way has operations \(\mu ^n_m : {V}^{\otimes m} \rightarrow {V}^{\otimes n}\), \(m,n \in \mathbb{N }\), \((m,n) \not = (1,1)\), but axioms as explicit as the ones for \(A_\infty \)-algebras were given only for \(m+n \le 6\).

1.2 Pitfalls

It is clearly desirable to have some polyhedral PROP \({ KK}= \{{ KK}^n_m\}\) playing the same rôle for \(A_\infty \)-bialgebras as the Stasheff’s operad plays for \(A_\infty \)-algebras. By this we mean that \(\mathcal{B }_\infty \) should be isomorphic to the PROP of cellular chains of \({ KK}\), so the differential in \(\mathcal{B }_\infty \) and therefore also the axioms of \(A_\infty \)-bialgebras would be encoded in the combinatorics of \({ KK}\) . To see where the pitfalls are hidden, we try to mimic the inductive construction of the associahedra in the context of bialgebras.

The first step is obvious. Assume we have a dg-vector space \(V\) with a multiplication \(\mu : {V}^{\otimes 2} \rightarrow V\) and a comultiplication \(\Delta : V \rightarrow {V}^{\otimes 2}\) such that \(\mu \) is associative up to a homotopy \(\mu _3^1 : {V}^{\otimes 3} \rightarrow V\) symbolized by the interval \(\mu \) and \(\Delta \) are compatible up to a homotopy \(\mu ^2_2 : {V}^{\otimes 2} \rightarrow {V}^{\otimes 2}\) symbolized byand \(\Delta \) is coassociative up to a homotopy \(\mu _1^3 : V \rightarrow {V}^{\otimes 3}\) depicted asLet us take all elements of \({V}^{\otimes 2}\) constructed out of three elements of \(V\) using \(\Delta \) and the multiplication on the tensor powers of \(V \) induced in the standard manner by \(\mu \). Let us call such elements algebraic. There are six of them, labelling the vertices of a hexagon:All products labelling adjacent vertices except the two bottom ones are homotopic via an ‘algebraic’ homotopy, i.e. a homotopy constructed using \(\Delta \), \(\mu ^1_3\), \(\mu ^2_2\), and the multiplication induced by \(\mu \) on the powers of \(V\).
Let us inspect the vertices \(L\) and \(R\). The ‘obvious’ candidate \(\mu ^1_3\left( \Delta (a),\Delta (b),\Delta (c)\right) \) for the connecting homotopy does not have any meaning. The labels of these vertices are, however, still homotopic but in an unexpected manner. For \(a,b,c \in V\) define \(X(a,b,c) \in {V}^{\otimes 2}\) by
$$\begin{aligned} X(a,b,c):= \left( \mu (1\!\!1\otimes \mu ) \otimes \mu (\mu \otimes 1\!\!1)\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) \end{aligned}$$
where \(\sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) : {V}^{\otimes 6}\rightarrow {V}^{\otimes 6}\) is the permutation acting on \(v_1,\ldots ,v_6 \in V\) as
$$\begin{aligned} \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) (v_1 \otimes v_2 \otimes v_3 \otimes v_4 \otimes v_5 \otimes v_6):= (v_1 \otimes v_3 \otimes v_5 \otimes v_2 \otimes v_4 \otimes v_6). \end{aligned}$$
Similarly, put
$$\begin{aligned} Y(a,b,c):= \left( \mu (\mu \otimes 1\!\!1) \otimes \mu (1\!\!1\otimes \mu )\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) . \end{aligned}$$
Define furthermore the homotopies \(H_l,H_r,G_l,G_r : {V}^{\otimes 3} \rightarrow {V}^{\otimes 2}\) by the formulas
$$\begin{aligned} H_l(a,b,c)&:= \left( \mu ^1_3 \otimes \mu (\mu \otimes 1\!\!1)\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) ,\\ H_r(a,b,c)&:= \left( \mu (1\!\!1\otimes \mu ) \otimes \mu ^1_3\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) ,\\ G_l(a,b,c)&:= \left( \mu (\mu \otimes 1\!\!1) \otimes \mu ^1_3\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) , \ \hbox { and }\\ G_r(a,b,c)&:= \left( \mu ^1_3\otimes \mu (1\!\!1\otimes \mu )\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) . \end{aligned}$$
Observing that
$$\begin{aligned} \left( \Delta (a)\Delta (b)\right) \Delta (c) \!&= \! \left( \mu (\mu \otimes 1\!\!1) \otimes \mu (\mu \otimes 1\!\!1)\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)4\right) , \hbox { and }\\ \Delta (a)\left( \Delta (b)\Delta (c)\right)&= \left( \mu (1\!\!1\otimes \mu )\otimes \mu (1\!\!1\otimes \mu )\right) \sigma \left( \begin{array}{c} 3\\ 2 \end{array} \right) \left( \Delta (a) \otimes \Delta (b) \otimes \Delta (c)\right) , \end{aligned}$$
we see the following composite chain of homotopiesand alsoTo proceed as in the case of the associahedron, we need to subdivide the bottom edge of the hexagon \(K^2_3\) in (1) and consider the heptagon \({ KK}^2_3\) Observe that the subdivision and therefore also \({ KK}^2_3\) is not unique, we could as well take \(Y,G_l,G_r\) instead of \(X,H_l,H_r\). Notice also that neither the expressions \(X\), \(Y\) nor the homotopies \(H_l,H_r,G_l,G_r\) are algebraic.

1.3 Two types of biassociahedra

We can already glimpse the following pattern. There naturally appear polytopes \(K^n_m\), \(m,n \in \mathbb{N }\), such that \(K^1_m\) and \(K^m_1\) are isomorphic to Stasheff’s associahedron \(K_m\). We will also see that \(K^2_m\) is isomorphic to the multiplihedron \(J_m\) [2, 12]. We call these polytopes the step-one biassociahedra. In this paper we give a simple and clean description of their face posets.

To continue as in the case of \(A_\infty \)-algebras, one however needs to subdivide some faces of \(K^n_m\); and an example of such a subdivision is the heptagon \({ KK}^2_3\) in (2) subdividing the hexagon \(K^2_3\). The subdivisions must be compatible so that the result will be a cellular PROP \({ KK}= \{{ KK}^n_m\}\). Its associated cellular chain complex is moreover required to be isomorphic to the minimal model \(\mathcal{B }_\infty \) of the bialgebra PROP. We call these polyhedra step-two biassociahedra.

The polytopes \({ KK}^n_m\) were, for \(m+n \le 6\), constructed in [7]. In higher dimensions, the issue of the compatibility of the subdivisions arises. In [10], a construction of the step-two biassociahedra was proposed, but we admit that we were not able to verify it. In our opinion, a reasonably simple construction of the polyhedral PROP \({ KK}\) or at least a convincing proof of its existence still remains a challenge.

We think that a necessary starting point to address the above problems is a suitable notation. In this paper we give a simple description of the face poset of the step-one biassociahedron \(K^n_m\) that generalize the classical description of the Stasheff associahedron in terms of planar directed trees. We will also give a ‘coordinate-free’ characterization of \(K^n_m\) which shows that it is not a human invention but has existed since the beginning of time. As a by-product of our approach, it will be obvious that \(K^2_m\) is isomorphic to the multiplihedron \(J_m\), for each \(m \ge 2\).

1.4 Notation and terminology

Some low-dimensional examples of the step-two biassociahedra appeared for the first time, without explicit name, in [7]; they were denoted \(B^n_m\) there. The word biassociahedron was used by Saneblidze and Umble, see e.g. [10], referring to what we called above the step-two biassociahedron; they denoted it \({ KK}_{m,n}\). Step-one biassociahedra can also be, without explicit name, found in [10]; they were denoted \(K_{m,n}\) there. Whenever we mention the biassociahedron in this paper, we always mean the step-one biassociahedron which we denote by \(K^n_m\).

2 Main results

Let us recall some standard facts [9, 14]. The permutahedron 3 \(P_{m-1}\) is, for \(m \ge 2\), a convex polytope whose poset of faces \(\mathcal{P }_{m-1}\) is isomorphic to the set \({ lT}_m\) of planar directed trees with levels and \(m\) leaves, with the partial order generated by identifying adjacent levels. The permutahedron \(P_{m-1}\) can be realized as the convex hull of the vectors obtained by permuting the coordinates of \((1,\ldots ,m-1) \in \mathbb{R }^{m-1}\), its vertices correspond to elements of the symmetric group \(\Sigma _{m-1}\). The face poset \(\mathcal{K }_m\) of the Stasheff’s associahedron \(K_m\) is the set of directed planar trees (no levels) \(T_m\) with \(m\) leaves; the partial order is given by contracting the internal edges. The obvious epimorphism \(\varpi _m : lT_m \twoheadrightarrow T_m\) erasing the levels induces the Tonks projection \({ Ton}: \mathcal{P }_{m-1} \twoheadrightarrow \mathcal{K }_m\) of the face posets, see [14] for details.

There is a conceptual explanation of Tonks’ projection that uses a natural map
$$\begin{aligned} \varpi _m : { lT}_m \rightarrow \mathsf{F}(\xi _2,\xi _3,\ldots )(m) \end{aligned}$$
(3)
to the arity \(m\) piece of the free non-\(\Sigma \) operad [8, Section 4] generated by the operations \(\xi _2,\xi _3,\ldots \) of arities \(2,3,\ldots \), respectively. The map \(\varpi _m\), roughly speaking, replaces the vertices of a tree \(T \in { lT}_m\) with the generators of \(\mathsf{F}(\xi _2,\xi _3,\ldots )\) whose arities equal the number of inputs of the corresponding vertex, and then composes these generators using \(T\) as the composition scheme, see Sect. 4.1.
It is almost evident that the set \(T_m\) is isomorphic to the image of \(\varpi _m\). In other words, the face poset \(\mathcal{K }_m\) of the associahedron \(K_m\) can be defined as the quotient of \(lT_{m}\) modulo the equivalence that identifies elements having the same image under \(\varpi _m\), with the induced partial order. Tonks’ projection then appears as the epimorphism in the factorizationThe aim of this note is to define in the same manner the poset \(\mathcal{K }^n_m\) of faces of the step-one biassociahedron \(K^n_m\) constructed in [10, §9.5].4 To this end, we introduce in 3.2, for each \(m,n \ge 1\), the set \({ lT}^n_m\) of complementary pairs of directed planar trees with levels. The set \({ lT}^n_m\) has a partial order \(<\) similar to that of \({ lT}_m\). The poset \(\mathcal{P }^n_m = ({ lT}^n_m,<)\) provides a natural indexing of the face poset of the bipermutahedron \(P^n_m\) of [10].5 It turns out that the posets \(\mathcal{P }^n_m\) and \(\mathcal{P }_{m+n-1}\) are isomorphic; we give a simple proof of this fact in Sect. 3. Comparing it with the proof of the analogous [10, Proposition 6] convincingly demonstrates the naturality of our language of complementary pairs.
As the next step, we describe, for each \(m,n \ge 1\), a natural map
$$\begin{aligned} \varpi ^n_m : { lT}^n_m \rightarrow \mathsf{F}\left( \xi ^b_a \, | \, a,b \ge 1,\ (a,b) \not = (1,1)\right) . \end{aligned}$$
(4)
The object in the right hand side is the free PROP [8, Section 8] generated by operations \(\xi ^b_a\) of biarity \((b,a)\), i.e. with \(b\) outputs and \(a\) inputs. We then define the face poset \(\mathcal{K }^n_m\) of the biassociahedron as \({ lT}^n_m\) modulo the relation that identifies the complementary pairs of trees having the same image under \(\varpi ^n_m\), with the induced partial order. We prove that \(\mathcal{K }^n_m\) is isomorphic to the poset of complementary pairs of planar directed trees \({ zT}^n_m\) with zones, see Definition A. It will be obvious that this is the simplest possible description of the poset of faces of the Saneblidze–Umble biassociahedron that generalizes the standard description of the face poset of the Stasheff’s associahedron.

In the last section, we analyze in detail the special case of \(\mathcal{K }^2_m\) when complementary pairs of trees with zones can equivalently be described as trees with a diaphragm. Using this description we prove that \(\mathcal{K }^2_m\) is isomorphic to the face poset of the multiplihedron \(J_m\). Necessary facts about PROPs and calculus of fractions are recalled in the Appendix.

The main definitions are Definition A and Definition B. The main result is Theorem C and the main application is Proposition D.

3 Trees with levels and (bi)permutahedra

3.1 Up- and down-rooted trees

Let us start by recalling some classical material from [9, II.1.5]. A planar directed (also called rooted) tree is a planar tree with a specified leg called the root. The remaining legs are the leaves. We will tacitly assume that all vertices have at least three adjacent edges.

We will distinguish between up-rooted trees all of whose edges different from the root are oriented towards the root while, in down-rooted trees, we orient the edges to point away from the root. The set of vertices \({ Vert}(T)\) of an up- or down-rooted tree \(T\) is partially ordered by requiring that \(u < v\) if and only if there exist an oriented edge path starting at \(u\) and ending at \(v\).

An up-rooted planar tree with \(h\) levels, \(h \ge 1\), is an up-rooted planar tree \(U\) with vertices placed at \(h\) horizontal lines numbered \(1,\ldots ,h\) from the top down. More formally, an up-rooted tree with \(h\) levels is an up-rooted planar tree \(U\) together with a strictly order-preserving level function \(\ell : { Vert}(U) \rightarrow \{1,\ldots ,h\}\).6 We tacitly assume that the level function is an epimorphism (no ‘dummy’ levels with no vertices); if this is not the case, we say that \(\ell \) is degenerate. We believe that Fig. 1 clarifies these notions. Since we numbered the level lines from the top down, saying that vertex \(v'\) lies above \(v''\) means \(\ell (v') < \ell (v'')\).
Fig. 1

An up-rooted tree with ten leaves and five vertices aligned at four levels, none of them ‘dummy.’ The edges are oriented towards the root

Let us denote by \({ lT}_m\) the set of up-rooted trees with levels and \(m\) leaves. It forms a category whose morphisms \((U',\ell ') \rightarrow (U'',\ell '')\) are couples \((\phi ,\hat{\phi })\) consisting of a map7 of up-rooted planar trees \(\phi : U' \rightarrow U''\) and of an order-preserving map \(\hat{\phi }: \{1,\ldots ,h'\} \rightarrow \{1,\ldots ,h''\}\) forming the commutative diagramin which we denote \(\phi : U' \rightarrow U''\) and the induced map \({ Vert}(U) \rightarrow { Vert}(U'')\) by the same symbol.
We say that \((U',\ell ') < (U'',\ell '')\) if there exists a morphism \((U',\ell ') \rightarrow (U'',\ell '')\). Since all endomorphisms in \({ lT}_m\) are the identities, the relation \(<\) is a partial order. The set \({ lT}_m\) with this partial order is isomorphic to the face poset \(\mathcal{P }_{m-1}\) of the permutahedron \(P_{m-1}\). This result is so classical that we will not give full details here, see [14].8 For \(m=4\), this isomorphism is illustrated in Fig. 2.
Fig. 2

The faces of the permutahedron \(P_3\) indexed by the set \({ lT}_4\). The ordered partitions of \(\{1,2,3\}\) corresponding to the faces under the correspondence described in Sect. 3.3 are also shown

The definition of the poset \(({ lT}^n,<)\) of down-rooted trees with levels and \(n\) leaves is similar. We will also need the exceptional tree Open image in new window with one edge and no vertices. We define Open image in new window .

3.2 Complementary pairs

For \(m,n \ge 1\) we denote by \({ lT}^n_m\) the set of all triples \((U,D,\ell )\) consisting of an up-rooted planar tree \(U\) with \(m\) leaves and a down-rooted planar tree \(D\) with \(n\) leaves, equipped with a strictly order-preserving level function
$$\begin{aligned} \ell : { Vert}(U) \cup { Vert}(D) \twoheadrightarrow \{1,\ldots ,h\}. \end{aligned}$$
(5)
Observe that if we denote \(\ell _u:= \ell |_{{ Vert}(U)}\) (resp. \(\ell _d:= \ell |_{{ Vert}(D)}\)), then \((U,\ell _u)\) (resp. \((D,\ell _d)\)) is an up-rooted (resp. down-rooted) rooted tree with possibly degenerate level function.
We call the objects \((U,D,\ell )\) the complementary pairs of trees with levels. Figure 3 explains the terminology, concrete examples can be found in Fig. 6. When the level function is clear from the context, we drop it from the notation. The set \({ lT}^n_m\) forms a category in the same way as \({ lT}_m\). A morphism
$$\begin{aligned} \phi : (U',D',\ell ') \rightarrow (U'',D'',\ell '') \end{aligned}$$
is a triple \((\phi _u,\phi _d,\hat{\phi })\) consisting of a morphism \(\phi _u : U' \rightarrow U''\) (resp. \(\phi _d : D' \rightarrow D''\)) of up-rooted (resp. down-rooted) planar trees and of an order-preserving map \(\hat{\phi }: \{1,\ldots ,h'\} \rightarrow \{1,\ldots ,h''\}\) such that the diagramcommutes. The partial order of \({ lT}^n_m\), analogous to that of  \({ lT}_m\), is given by the existence of a morphism in the above category.
Fig. 3

A schematic picture of a couple \((U,D) \in { lT}^n_m\) of complementary trees with four levels

Theorem 3.1

The posets \(\mathcal{P }_{m+n-2} = ({ lT}_{m+n-1},<)\) and \(\mathcal{P }^n_m = ({ lT}^n_m,<)\) are naturally isomorphic for each \(m,n \ge 1\).

Proof

Let us describe an isomorphism of the underlying sets. Since clearly \({ lT}^1_u \cong { lT}^u_1 \cong { lT}_u\) for each \(u\ge 1\), it is enough to construct, for each \(n\ge 2\), \(m \ge 1\), an isomorphism
$$\begin{aligned} { lT}^n_m \cong { lT}^{n-1}_{m+1}. \end{aligned}$$
(7)
Let \(X = (U,D') \in { lT}^n_m\). Denote by \(v\) the initial vertex of the leftmost leaf of \(D'\) and \(L\) its level line. Amputation of this leftmost leaf at \(v\) gives a down-rooted tree \(D\) with \(n-1\) leaves. Now extend the up-rooted tree \(U\) by grafting an up-going leaf at the rightmost point \(w\) in which the level line \(L\) intersects \(U\). If \(w\) is a vertex, we graft the leaf at this vertex, if \(w\) is a point of an edge, we introduce a new vertex with two input edges.
Let \(U'\) denotes this extended tree. The isomorphism (7) is given by the correspondence \((U,D') \mapsto (U',D)\), with the pair \((U',D)\) equipped with the level function induced, in the obvious manner, by the level function of \((U,D')\). We believe that Fig. 4 makes isomorphism (7) obvious. It would, of course, be possible to define it using the formal language of trees with level functions, but we consider the above informal, intuitive definition more satisfactory.
Fig. 4

Isomorphism (7)

It is simple to verify that (7) preserves the partial orders, giving rise to a poset isomorphism \(\mathcal{P }^n_m \cong \mathcal{P }^{n-1}_{m+1}\), for each \(n\ge 2\), \(m \ge 1\). \(\square \)

Example

The isomorphism of Theorem 3.1 is, for \(m+n = 4\), presented in Fig. 5.

Fig. 5

The isomorphic posets \(\mathcal{P }_2 = \mathcal{P }^1_3\), \(\mathcal{P }^2_2\) and \(\mathcal{P }^3_1\)

Example

Isomorphism (7) of complementary pairs is, for \(m\,+\,n=5\), shown in the table of Fig. 6. Comparing the entries in the leftmost column with the corresponding entries of the 4th one, we see a nontrivial isomorphism between the posets of up- and down-rooted trees. We suggest, as an exercise, to decorate the faces of the permutahedron \(P_4\) in Fig. 2 by the corresponding entries of the table in Fig. 6 to verify in this particular case that the partial orders are indeed preserved.
Fig. 6

The isomorphic sets \({ lT}^n_m\) with \(m+n = 5\), \(\varpi ({ lT}^2_3)\), \({ zT}^2_3\) and \({ pT}_3\). The upper section of the left part corresponds to the vertices, the middle to the edges, and the bottom row to the 2-cell of the bipermutahedron \(P_3 = P^1_4 \cong P^2_3 \cong P^3_2 \cong P^4_1\)

3.3 Relation to the standard permutahedron

In this subsection we recall the well-known isomorphism between the poset \(\mathcal{P }_m = ({ lT}_m,<)\) of rooted planar trees with levels and the poset of ordered decompositions of the set \(\{1,\ldots ,m-1\}\) which we denote by \(\mathcal{SP }_m = ({ Dec}_m,<)\) (the \(\mathcal{S }\) in front of \(\mathcal{P }\) abbreviating the standard permutahedron). This isomorphism, which forms a necessary link to [10] and related papers, extends to an isomorphism between \((\mathcal{P }^n_m,<)\) and the poset of ordered bipartitions \(\mathcal{SP }^n_m = ({ Dec}^n_m,<)\). As the gadgets described here will not be used later in our note, this subsection can be safely skipped.

We start by drawing a tree \(U \in { lT}_m\), as always in this note, with the root up, and labelling the intervals between its leaves, from the left to the right, by \(1,\ldots ,m-1\). We then replace the labels by party balloons, release them and let them lift to the highest possible level.9 The first set of the corresponding partition is formed by the balloons that lifted to the root vertex (level \(1\)), the second by the balloons that lifted to level \(2\), &c. For instance, to the tree \(U \in { lT}_8\) in Fig. 7 one associates the ordered decomposition \((4|57|12|36) \in { Dec}_8\). Another instance of the above correspondence is shown in Fig. 2.
Fig. 7

An up-rooted tree \(U \in { lT}_8\) with party balloons ready to take off

Let us denote the resulting isomorphism by \(\gamma : \mathcal{P }_m \cong \mathcal{SP }_m\). Combined with the isomorphisms of Theorem 3.1, it leads to an isomorphism (denoted by the same symbol) \(\gamma : \mathcal{P }^n_m \cong \mathcal{SP }_{m+m-2}\) for each \(m,n \ge 1\).

Example

A particular instance of the above isomorphism is \(\gamma : \mathcal{P }^m \cong \mathcal{SP }_m\). On the other hand, the posets \(\mathcal{P }_m = ({ lT}_m,<)\) and \(\mathcal{P }^m = ({ lT}^m,<)\) are in fact the same, it is only that we draw the trees in \(\mathcal{P }_m\) with the root up, and those in \(\mathcal{P }^m\) with the root down. One can prove that the composite
$$\begin{aligned} \tau : \mathcal{SP }_m \stackrel{\gamma ^{-1}}{\cong }\mathcal{P }_m = \mathcal{P }^m \stackrel{\gamma }{\cong }\mathcal{SP }_m \end{aligned}$$
(8)
is given by reversing the order of the members of the decomposition. For instance,
$$\begin{aligned} \tau (4|57|12|36) = (36|12|57|4). \end{aligned}$$
Let us extend the above isomorphism to complementary pairs of trees. For   \(m,n \ge 1\), denote by \(\mathcal{SP }^n_m = ({ Dec}^n_m,<)\) the poset of ordered bipartitions of the sets \(\{1,\ldots ,m-1\}\), \(\{m,\ldots ,m+n-2\}\), by which we mean arrayswhere \(U_1,U_2,\ldots ,U_\ell \) (resp. \(D_1,D_2,\ldots ,D_\ell )\) is an ordered partition of \(1,\ldots ,m-1\) (resp. an ordered partition of \(m,\ldots , m+n-2\)). Here we allow some of the sets \(U_j\) (resp. \(D_j\)) to be empty, but we require \(U_j \cup D_j \not = \emptyset \) for each \(1\le j \le \ell \).
We associate to a complementary pair \(X = (U,D) \in { lT}^n_m\) a bipartition in \({ Dec}^n_m\) as follows. We attach to the intervals between the leaves of the up-rooted tree \(U\) balloons labeled \(1,\ldots ,m-1\), and to the intervals between the leaves of the down-rooted tree \(D\) balls labelled \(m,\ldots ,m+n-2\) as indicated in Fig. 8.
Fig. 8

A complementary pair decorated by \(m+n-2\) balloons and balls. The labels \(m-1\), \(m+1\) and \(n+m-2\) are not shown since the balloons resp. balls are too small to hold them

Then \(U_j\) (resp. \(D_j\)) is the set of balloons (resp. balls) that lift (resp. fall) to level \(j\), \(1 \le j \le \ell \). Observe that the reversing map (8) is already built in the above assignment.

Example

The following bipartitions correspond to the entries of the second line of the first part of the table in Fig. 6:while the bipartitions corresponding to the second line of the second part are

As an exercise, we recommend describing the isomorphism of Theorem 3.1 in terms of bipartitions. The above example serves as a clue how to do so.

3.4 Relation to strings of matrices

Saneblidze and Umble pointed out in [10] that the vertices of the bipermutahedron \(\mathcal{P }^n_m\) (denoted \(P_{m-1,n-1}\) in loc. cit.) can be indexed by strings \(E_1 \cdots E_h\) of matrices with constant rows and columns and entries from either Open image in new window or Open image in new window , satisfying certain combinatorial conditions which we list at the end of this subsection. On the other hand, the vertices of the bipermutahedron are the minimal elements of the poset \(({ lT}^n_m,<)\) of complementary pairs of trees by Sect. 3.3. We therefore have the isomorphisms
$$\begin{aligned} \{\hbox {minimal elements of }{ lT}^n_m\} \stackrel{\cong }{\longrightarrow }\{\hbox {vertices of }\mathcal{P }^n_m\} \stackrel{\cong }{\longrightarrow }\{\hbox {strings } E_1\cdots E_h\}. \end{aligned}$$
In this subsection we describe the composite isomorphism
$$\begin{aligned} { str}: \{\hbox {minimal elements of }{ lT}^n_m\} \stackrel{\cong }{\longrightarrow }\{\hbox {strings } E_1\cdots E_h\}. \end{aligned}$$
In Sect. 5.3 we indicate how to use this description to prove that the step-one biassociahedron \(\mathcal{K }^n_m\) constructed in Sect. 4 is isomorphic to the face poset of the polytope \(K_{m-1,n-1}\) of [10]. We will not use the material presented here in other parts of this article, so this subsection can be skipped.
We will consider matrices with three types of entries, Open image in new window and Open image in new window , symbolizing multiplication, comultiplication and the identity. We will distinguish two types of these matrices.

Definition

A Open image in new window -matrix (resp.  a Open image in new window -matrix) is elementary if it contains precisely one column (resp. row) of Open image in new window ’s (resp.  Open image in new window ’s).

Example

The first two matrices below are elementary, the middle one is a non-elementary Open image in new window -matrix, the remaining two matrices are not of the required type:

We are going to describe how a minimal element \(X =(U,D,\ell )\) of the poset \(({ lT}^n_m,<)\) determines a string \({ str}(X)\) of elementary matrices. Minimality of \(X\) means that each level is occupied by precisely one vertex, which is either Open image in new window or Open image in new window . In particular, the level function (5) is an isomorphism. As before we denote by \(\ell _u\) (resp. \(\ell _d\)) its restriction to \({ Vert}(U)\) (resp. \({ Vert}(D)\)).

The up-rooted part \(U\) of \(X\) is a tree with \(s\) levels, where \(s\) is the cardinality of \(\mathrm{Im}(\ell _u)\). Likewise, the down-rooted part \(D\) is a tree with \(t\) levels, where \(t\) is the cardinality of \(\mathrm{Im}(\ell _d)\). Clearly \(s+t = h\), \(m=s+1\) and \(n= t+1\). Write
$$\begin{aligned} \mathrm{Im}(\ell _u) = \{u_1 < \cdots < u_s\} \quad \hbox { and } \quad \mathrm{Im}(\ell _d) = \{d_1 < \cdots < d_t\} . \end{aligned}$$
The string of elementary matrices corresponding to \(X\) will be of the form \(E_1 \cdots E_h\), where \(E_i\) is an elementary Open image in new window -matrix if \(i \in \mathrm{Im}(\ell _u)\) and an elementary Open image in new window -matrix if \(i \in \mathrm{Im}(\ell _d)\). So we distinguish two cases.
  • The matrices \(E_{u_a}\), \(1 \le a \le s\). Let \(L\) be the \(a\)th level line of \(U\), i.e. the \(u_a\)th level line of \(X\). Its intersection with \(U\) determines a vector \([R_a]\) whose entries represent the points in \(L \cap U\) read from left to right. An intersection with an edge is represented by Open image in new window , the (unique) intersection in a vertex by Open image in new window . It is clear that the sequence \([R_1],\ldots ,[R_s]\) determines \(U\).

Example

The above construction associates to Open image in new window the sequence Open image in new window . Organizing the vectors \([R_1],\ldots ,[R_s]\) into a vertical array elucidates their meaning; for the above tree we get
The matrix \(E_{u_a}\) is, as each elementary Open image in new window -matrix, determined by one of its rows (all its rows are the same) and the number of its rows. We postulate that the row of \(E_{u_a}\) is \([R_a]\). The number of its rows is defined as
$$\begin{aligned} \mathrm{card}\left\{ i \in \{1,\ldots ,h\}\ |\ i > u_a,\ i \in \mathrm{Im}(\ell _d)\right\} +1. \end{aligned}$$
  • The matrices \(E_{d_b}\), \(1 \le b \le t\). We proceed analogously to the previous case. To the \(b\)th level line \(L\) of \(D\), i.e. the \(d_b\)th level line of \(X\), we associate a ‘vertical vector’ (a column) \(C_b\) whose entries are determined by the intersection \(L \cap D\) as in the previous case. The only difference is that the (unique) intersection with a vertex is now represented by Open image in new window . The column of \(E_{d_b}\) is \(C_b\) and the number of its columns is
    $$\begin{aligned} \mathrm{card}\left\{ i \in \{1,\ldots ,h\}\ |\ i < u_a,\ i \in \mathrm{Im}(\ell _u)\right\} +1. \end{aligned}$$

Example

The strings corresponding to complementary pairs in the third row of the table in Fig. 6 are:In Sect. 5.3 we refer also to the following two strings of elementary matrices corresponding to complementary pairs in \({ lT}^2_4\):
We associated to each minimal element \(X \in { lT}^n_m\) a string \({ str}(X) = E_1\cdots E_h\) of elementary matrices. While it is clear from the construction that the map \({ str}\) is a monomorphism, not all products of elementary matrices are in \(\mathrm{Im}({ str})\). It however follows from the methods of [10] that \(E_1\cdots E_h \in { str}({ lT}^n_m)\) if and only if
  1. (i)

    \(E_1\cdots E_h\) contains precisely \(m-1\) Open image in new window -matrices and \(n-1\) Open image in new window -matrices,10

     
  2. (ii)

    the leftmost Open image in new window -matrix has one column, the rightmost Open image in new window -matrix has one row, and

     
  3. (iii)

    \(E_1\cdots E_h\) consist of block transverse pairs.

     
While the meaning of (i) and (ii) is clear, the block transversality of (iii) is a certain combinatorial property of pairs of matrices introduced in [10, §4.1]. For elementary matrices, it is equivalent to the following condition.
Let \({ col}(M)\) denote the number of columns of a matrix \(M\) and \({ row}(M)\) the number of its rows. Then, for each \(1 \le i < h\),
  • if both \(E_i\) and \(E_{i+1}\) are Open image in new window -matrices, then
    $$\begin{aligned} { row}(E_i) = { row}(E_{i+1}), { col}(E_i) +1 = { col}(E_{i+1}), \end{aligned}$$
  • if both \(E_i\) and \(E_{i+1}\) are Open image in new window -matrices, then
    $$\begin{aligned} { row}(E_i) = { row}(E_{i+1}) +1, { col}(E_i) = { col}(E_{i+1}), \end{aligned}$$
  • if \(E_i\) is a Open image in new window -matrix and \(E_{i+1}\) a Open image in new window -matrix, then
    $$\begin{aligned} { row}(E_i) = { row}(E_{i+1}) +1, { col}(E_i) +1 = { col}(E_{i+1}), \end{aligned}$$
  • if \(E_i\) is a Open image in new window -matrix and \(E_{i+1}\) a Open image in new window -matrix, then
    $$\begin{aligned} { row}(E_i) = { row}(E_{i+1}), { col}(E_i) = { col}(E_{i+1}). \end{aligned}$$
We leave as an exercise to check that the string \({ str}(X)\) constructed above indeed satisfies the conditions (i)–(iii).

4 Trees with zones and the biassociahedron \(\mathcal{K }^n_m\)

In this section we present our definition of the face poset \(\mathcal{K }^n_m\) of the biassociahedron. Let us recall the classical associahedron first.

4.1 The associahedron \(K_m\)

As in (3), denote by \(\mathsf{F}(\xi _2,\xi _3,\ldots )\) the free non-\(\Sigma \) operad in the monoidal category of sets, generated by the operations of \(\xi _2,\xi _3,\ldots \) of arities \(2,3,\ldots \), respectively. Its component of arity \(n\) consists of (up-rooted) planar rooted trees with vertices having at least \(2\) inputs [8, Section 4]. We can therefore define the map (3) simply by forgetting the level functions. We however give a more formal, inductive definition which exhibits some features of other constructions used later in this note.

Let \((U,\ell ) \in { lT}_m\). Since our description of the map (3) will not depend on the level function, we drop it from the notation. If \(U\) is the up-rooted \(n\)-corolla \(c_m\), \(m \ge 2\), i.e. the tree with one vertex and \(m\) leaves, we put
$$\begin{aligned} \varpi (c_m) := \xi _m \in \mathsf{F}(\xi _2,\xi _3,\ldots )(m). \end{aligned}$$
Agreeing that \(c_1\) denotes the exceptional tree Open image in new window , we extend the above formula for \(m=1\) by
$$\begin{aligned} \varpi _1(c_1) := e \in \mathsf{F}(\xi _2,\xi _3,\ldots )(1), \end{aligned}$$
where \(e\) is the operad unit. Let us proceed by induction on the number of vertices. An arbitrary \(U \in { lT}_m\), \(m \ge 2\), is of the formwith some up-rooted, possibly exceptional, trees \(U_1,\ldots ,U_a\), \(a \ge 2\), each having strictly fewer vertices than \(U\). We then put
$$\begin{aligned} \varpi (U) := \xi _a\left( \varpi (U_1),\ldots ,\varpi (U_a)\right) \in \mathsf{F}(\xi _2,\xi _3,\ldots )(m), \end{aligned}$$
where \(-(-,\cdots ,-)\) in the right hand side denotes the operad composition. Notice that we simplified the notation by dropping the subscripts of \(\varpi \).
For \((U',\ell ), (U'',\ell '') \in { lT}_m\) let \((U',\ell ') \sim (U'',\ell '')\) if \(\varpi (U',\ell ') = \varpi (U'',\ell '')\). Since obviously the latter equality holds if and only if \(U' = U''\), the levels disappear and the quotient \({ lT}_m/\!\sim \) is isomorphic to the set \(T_m\) of up-rooted trees with \(m\) leaves. The partial order of \({ lT}_m\) induces the standard partial order11 on \(T_m\), so we have the isomorphism
$$\begin{aligned} \mathcal{K }_m \cong \mathcal{P }_{m-1}/\sim . \end{aligned}$$
We can take the above equation as a definition of the face poset of the associahedron \(K_m\). The discrepancy between the indices (\(m\) versus \(m-1\)) is of historical origin.

4.2 Complementary pairs with zones

For \(m,n \ge 1\), consider a triple \((U,D,z)\) consisting of an up-rooted planar tree \(U\) with \(m\) leaves, a down-rooted planar tree \(D\) with \(n\) leaves, and an order preserving epimorphism
$$\begin{aligned} z: { Vert}(U) \cup { Vert}(D) \twoheadrightarrow \{1,\ldots ,l\}. \end{aligned}$$
(11)
We call \(i\in \{1,\ldots ,l\}\) such that \(z^{-1}(i)\) contains both a vertex of \(U\) and a vertex of \(D\)barrier. The remaining \(i\)’s are the zones of \(z\). If \(z^{-1}(i) \subset { Vert}(U)\) (resp. \(z^{-1}(i) \subset { Vert}(D)\)), we call \(i\) an up-zone (resp. down-zone).

Definition A

We call (11) a zone function if
  1. (i)

    \(z\) is strictly order-preserving on barriers and

     
  2. (ii)

    there are no adjacent zones of the same type.

     
We denote by \({ zT}^n_m\) the set of all triples \((U,D,z)\) consisting of a planar up-rooted tree \(U\) with \(m\) leaves, a planar down-rooted tree with \(n\) leaves, and a zone function (11).
Condition (i) means the following. If \(u',u'' \in { Vert}(U)\) and \(v \in { Vert}(D)\) are such that \(z(u') = z(u'') = z(v)\), then \(u'\) and \(u''\) are unrelated.12 Dually, if \(v',v'' \in { Vert}(D)\) and \(u \in { Vert}(U)\) are such that \(z(u) = z(v') = z(v'')\), then \(v'\) and \(v''\) are unrelated. Condition (ii) can be rephrased as follows. For \(i \in (1,\ldots ,l)\) let
$$\begin{aligned} t_z(i) := \left\{ \begin{array}{ll} \mathrm{U}&{}\quad \hbox { if }i \hbox { is an up-zone},\\ \mathrm{D}&{}\quad \hbox { if }i \hbox { is an down-zone, and}\\ \mathrm{B}&{}\quad \hbox { if }i \hbox { is a barrier.} \end{array} \right. \end{aligned}$$
Condition (ii) then says that the sequence \(\left( t_z(1)\cdots t_z(l)\right) \) does not contains subsequences UU or DD. We call \(\left( t_z(1)\cdots t_z(l)\right) \) the type of \(z\).
The notion of complementary pairs with zones is illustrated in Fig. 9. In the picture, the values 1, 3, 4, 7 are zones, the values 2, 5, 6 are barriers. The type of the zone function is \((\mathrm{DBDUBBUB})\).
Fig. 9

An element of \({ zT}^n_m\)

Example

Let us look at case \(n=2\) of Definition A. For Open image in new window , only the following three cases may happen.

Case \(l = 1\). \(1\) is a barrier if \(m \ge 2\) and \(1\) is a down-zone if \(m=1\).

Case \(l = 2\). One has four possibilities for the type of \(z\), namely \((\mathrm{D}\mathrm{U}), (\mathrm{U}\mathrm{D}), (\mathrm{B}\mathrm{U})\) or \((\mathrm{U}\mathrm{B}).\) In the \((\mathrm{D}\mathrm{U})\) and \((\mathrm{U}\mathrm{D})\) cases \(m \ge 2\), in the remaining two cases \(m \ge 3\).

Case \(l = 3\). \(m \ge 3\) and the only possibility for the type is Alfred Jarry’s \((\mathrm{U}\mathrm{B}\mathrm{U})\) [3].

For \(i \in \{1,\ldots ,h\}\) define its closure \(\overline{i} \subset \{1,\ldots ,h\}\) by \(\overline{i} := \{i\}\) if \(i\) is a barrier, and \(\overline{i}\) to be the set consisting of \(i\) and its adjacent barriers in the opposite case. For the complementary pair in Fig. 9,
$$\begin{aligned} \begin{array}{llll} \overline{1} = \{1,2\}, &{} \overline{2} = \{2\}, &{} \overline{3} = \{2,3\}, &{}\overline{4} = \{4,5\},\\ \overline{5} = \{5\}, &{}\overline{6} = \{6\}, &{} \overline{7} = \{6,7,8\}, &{}\overline{8} = \{8\}. \end{array} \end{aligned}$$
A morphism \((U',D',z') \rightarrow (U'',D'',z'')\) is a triple \((\phi _u,\phi _d,\hat{\phi })\) consisting of morphisms \(\phi _u : U' \rightarrow U''\), \(\phi _d : D'' \rightarrow D''\) of trees and of an order-preserving map \(\hat{\phi }: \{1,\ldots ,l'\} \rightarrow \{1,\ldots ,l''\}\) such that the closures are preserved, that is
$$\begin{aligned} z''\left( \phi _u(u)\right) \in \overline{\hat{\phi }\left( z'(u)\right) } \quad \hbox { and } \quad z'' \left( \phi _d(v)\right) \in \overline{\hat{\phi }\left( z'(v)\right) }, \end{aligned}$$
for \(u \in { Vert}(U)\) and \(v \in { Vert}(D)\). We may also say that the obvious analog of (6), i.e.commutes up to the closures. The notion of a morphism induces a partial order \(<\).

Definition B

The face poset of the (step-one) biassociahedron is the poset \(\mathcal{K }^n_m := ({ zT}^n_m,<)\) of complementary pairs of trees with zones, with the above partial order.

Let \((U,D,\ell ) \in { lT}^n_m\) be a pair with the level function \(\ell : { Vert}(U) \cup { Vert}(D) \rightarrow \{1,\ldots ,h\}\). We call \(i \in \{1,\ldots ,h\}\) an up-level (resp. down-level) if \(\ell ^{-1}(i) \subset { Vert}(U)\) (resp. \(\ell ^{-1}(i) \subset { Vert}(D)\)).

Definition 4.1

Let \((U,D,\ell ) \in { lT}^n_m\) be as above and \((1,\ldots ,l)\) the quotient cardinal obtained from \((1,\ldots ,h)\) by identifying the adjacent up-levels and the adjacent down-levels. Denote by \(p: (1,\ldots ,h) \twoheadrightarrow (1,\ldots ,l)\) the projection and define
$$\begin{aligned} \pi (U,D,\ell ) := (U,D,z) \in { zT}^n_m, \ \hbox { with } z := p \circ \ell . \end{aligned}$$
We call the map \(\pi : { lT}^n_m \rightarrow { zT}^n_m\) defined in this way the canonical projection and \( z = p \circ \ell \) the induced zone function.

It is easy to show that the map \(\pi \) preserves the partial orders, giving rise to the projection \(\mathcal{P }^n_m \twoheadrightarrow \mathcal{K }^n_m\) of posets. We finish this subsection by two statements needed in the proof of Theorem C.

Proposition 4.2

Let \((U,D,z'), (U,D,z'') \in { zT}^n_m\). If, for each vertices \(u \in { Vert}(U)\) and \(v \in { Vert}(D)\),
$$\begin{aligned} z'(u) < z'(v)\ (\hbox {resp.}\ z'(u) = z'(v),\ \hbox {resp.~} z'(u) > z'(v)) \end{aligned}$$
(13a)
is equivalent to
$$\begin{aligned} z''(u) < z''(v)\ (\hbox {resp.}\ z''(u) = z''(v),\ \hbox {resp.} z''(u) > z''(v)), \end{aligned}$$
(13b)
then \(z' = z''\).

Proof

Let
$$\begin{aligned} z' : { Vert}(U) \cup { Vert}(D) \rightarrow \{1,\ldots ,h'\} \ \hbox { and } \ z'' : { Vert}(U) \cup { Vert}(D) \rightarrow \{1,\ldots ,h''\} \end{aligned}$$
be zone functions as in the proposition. Let us show that, for \(x,y \in { Vert}(U) \cup { Vert}(D)\),
$$\begin{aligned} \hbox {if }z'(x) = z'(y) \hbox { then } z''(x) = z''(y). \end{aligned}$$
(14)
The above implication immediately follows from the assumptions if \(x \in { Vert}(U)\) and \(y \in { Vert}(D)\), or if \(x \in { Vert}(D)\) and \(y \in { Vert}(U)\). So assume \(x,y \in { Vert}(U)\), \(z'(x) = z'(y)\) and, say, \(z''(x) > z''(y)\). Since, by definition, \(z''\) does not have two adjacent zones of type \(\mathrm{U}\), there must exist \(v \in { Vert}(D)\) such that \(z''(x) \ge z''(v) \ge z''(y)\), where at least one relation is sharp. Assuming the equivalence between (13a) and (13b), we get \(z'(x) \ge z'(v) \ge z'(y)\) where again at least one relation is sharp, so \(z'(x) \not = z'(y)\), which is a contradiction. The case \(x,y \in { Vert}(D)\) can be discussed in the same way, thus (14) is established.

For \(i \in \{1,\ldots ,h'\}\) (resp. \(j \in \{1,\ldots ,h''\}\)) denote \(S'_i := z'^{-1}(i)\) (resp. \(S''_j := z''^{-1}(j)\)). By (14), for each \(i\) there exists a unique \(j\) such that \(S'_i \subset S''_j\). Exchanging the rôles of \(z'\) and \(z''\), we see that, vice versa, for each \(j\) there exists a unique \(i\) such that \(S''_j \subset S'_i\). This obviously means that there exists an automorphism \(\varphi : \{1,\ldots ,h'\} \rightarrow \{1,\ldots ,h''\}\) such that \(z' = \varphi \circ z''\). As both \(z'\) and \(z''\) are order-preserving epimorphisms, \(\varphi \) must be the identity. \(\square \)

The following proposition in conjunction with Proposition 4.2 shows that the induced zone function remembers the relative heights of vertices of \(U\) and \(D\) but nothing more.

Proposition 4.3

Let \((U,D,\ell ) \in { lT}^n_m\) and \(z\) be the zone function induced by \(\ell \). Then, for each \(u \in { Vert}(U)\) and \(v \in { Vert}(D)\),
$$\begin{aligned} \ell (u) < \ell (v)\ (\hbox {resp.~}\ell (u) = \ell (v),\ \hbox {resp.~} \ell (u) > \ell (v)) \end{aligned}$$
if and only if
$$\begin{aligned} z(u) < z(v)\ (\hbox {resp.~}z(u) = z(v),\ \hbox {resp.~} z(u) > z(v)). \end{aligned}$$

Proof

The proof is a simple application of the definition of the induced zone function. \(\square \)

4.3 The map \(\varpi : { lT}^n_m \rightarrow \mathsf{F}(\Xi )\)

This subsection relies on the notation and terminology recalled in the Appendix. For \((U,D,\ell ) \in { lT}^n_m\) and subtrees \(\overline{U} \subset U\), \(\overline{D} \subset D\) with, say, \(\overline{m}\) and resp. \(\overline{n}\) leaves, one has a natural restriction
$$\begin{aligned} r_{\overline{U}, \overline{D}}(U,D,\ell ) = (\overline{U}, \overline{D}, \overline{\ell }) \in { lT}^{\overline{n}}_{\overline{m}} \end{aligned}$$
(15)
with the level function \(\overline{\ell }: { Vert}(\overline{U}) \cup { Vert}( \overline{D}) \rightarrow (1,\ldots ,\overline{h})\) defined as follows. The image of the restriction of \(\ell \) to \({ Vert}(\overline{U}) \cup { Vert}( \overline{D})\) is a sub-cardinal of \((1,\ldots , h)\), canonically isomorphic to \((1,\ldots ,\overline{h})\) for some \(\overline{h} \le h\). The level function \(\overline{\ell }\) is then the composition of the restriction of \(\ell \) with this canonical isomorphism. In other words, \(\overline{\ell }\) is the epimorphism in the factorization
$$\begin{aligned} { Vert}(\overline{U}) \cup { Vert}( \overline{D}) \stackrel{\overline{\ell }}{\twoheadrightarrow }(1,\ldots ,\overline{h}) \hookrightarrow (1,\ldots ,h) \end{aligned}$$
of the restriction of \(\ell \).
Let \(\mathsf{F}(\Xi )\) be the free PROP in the category of sets, generated by the bicollection
$$\begin{aligned} \Xi := \left\{ \xi ^n_m \ | \ m,n \ge 1, \ (m,n) \not = (1,1)\right\} , \end{aligned}$$
where \(\xi ^n_m\) denotes the generator of biarity \((n,m)\) (\(m\) inputs and \(n\) outputs). Observe that one has, for each \(m,n \ge 1\), the inclusions
$$\begin{aligned}&\iota _U: \mathsf{F}(\xi _2,\xi _3,\ldots )(m) \hookrightarrow \mathsf{F}(\Xi ) \left( \begin{array}{c} 1\\ m \end{array} \right) \ \hbox { and }\nonumber \\&\iota _D : \mathsf{F}(\xi _2,\xi _3,\ldots )(n) \hookrightarrow \mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ 1 \end{array} \right) , \end{aligned}$$
(16)
given by
$$\begin{aligned} \iota _U(\xi _a) := \xi ^1_a \ \hbox { resp. }\ \iota _D(\xi _a) := \xi _1^a,\ a \ge 2. \end{aligned}$$
We will use \(\iota _U\) to identify \(\mathsf{F}(\xi _2,\xi _3,\ldots )(m)\) with a subset of \(\mathsf{F}(\Xi ) \left( \begin{array}{c} 1\\ m \end{array} \right) \).
Let us start the actual construction of the map \(\varpi : { lT}^n_m \rightarrow \mathsf{F}(\Xi )\left( \begin{array}{l}m\\ n\end{array}\right) \). First of all, for a down-rooted tree \(D\), i.e. an element of \({ lT}^n_1\), define
$$\begin{aligned} \varpi (D) := \iota _D \left( \varpi (D')\right) \in \mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ 1 \end{array} \right) , \end{aligned}$$
where \(D'\) is \(D\) turned upside down (i.e. an up-rooted tree), \(\varpi \) is as in 4.1 and \(\iota _D\) the second inclusion of (16).

Each element \(X \in { lT}^n_m\) is a triple \(X = (U,D,\ell )\), where \(U\) is an up-rooted tree with \(m\) leaves and \(D\) a down-rooted tree with \(n\) leaves. We construct \(\varpi (X)\) by induction on the number of vertices of \(D\). We distinguish three cases.

Case 1. The root vertex of \(D\) is above the root vertex of \(U\),13 schematicallyThen we put
$$\begin{aligned} \varpi (X) := \frac{\varpi (D)}{\varpi (U)} = \varpi (D) \circ \varpi (U) \in \mathsf{F}(\Xi ) \left( \begin{array}{l}m\\ n\end{array}\right) . \end{aligned}$$
(17)
Case 2. The vertex of \(D\) is at the same level as the root vertex of \(U\). We decompose \(U\) as in (10) and \(D\) in the obvious dual manner, the result is portrayed in Fig. 10.
Fig. 10

The decomposition of \(U\) and \(D\) in the 2nd case

We then define Case 3. The root vertex of \(D\) is below the root vertex of \(U\). In this case we decompose \(U\) and \(D\) as in Fig. 11 in which \(T\) is the maximal up-rooted tree containing all vertices above the level of the root vertex of \(D\) and the up-rooted trees \(U_1,\ldots ,U_a\) contain all the remaining vertices of \(U\). The decomposition of \(D\) is the same as in Case 2. Using the restriction (15), we denote
Fig. 11

The decomposition of \(U\) and \(D\) in the 3rd case

$$\begin{aligned} X_i := r_{U_i,c^b_1}(X),\ 1\le i \le a,\ \hbox { and }\quad Y_j := r_{T,D_j}(X),\ 1\le j \le b. \end{aligned}$$
(19)
Clearly \(\varpi (X_i)\)’s fall into the previous two cases. Since \(D_j\) has strictly less vertices than \(D\), \(\varpi (Y_j)\)’s have been defined by induction. We put
$$\begin{aligned} \varpi (X) = \frac{\varpi (Y_1) \cdots \varpi (Y_b)}{\varpi (X_1)\cdots \varpi (X_a)} \in \mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) . \end{aligned}$$
(20)

Remark

In the above construction of the map \(\varpi \), the root vertex of \(D\) plays a different rôle than the root vertex of \(U\). One can exchange the rôles of \(U\) and \(D\), arriving at a formally different formula for \(\varpi (X)\). Due to the associativity of fractions [7, Section 6], both formulas give the same element of \(\mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) \). It is also possible to write a non-inductive formula for \(\varpi (X)\) based on the technique of block transversal matrices developed in [10].

Example

Ifwe are in Case 3, with \(T = U_1\) the up-rooted \(2\)-corollas Open image in new window , and \(U_2\) the exceptional tree Open image in new window . We haveBoth \(X_1\) and \(X_2\) fall into Case 1, and \(\varpi (X_1) = \xi _1^2 \circ \xi ^1_2\) while \(\varpi (X_2) = \xi ^2_1\). Formula (20) givesThe rightmost term is obtained by depicting \(\xi ^n_m\) as an oriented corolla with \(m\) inputs and \(n\) outputs. The same notation is used in the 5th column of the table of Fig. 6 which lists elements in the image \(\varpi ({ lT}^2_3)\).

Let us formulate the main result of this note which relates the canonical projection of Definition 4.1 with the map \(\varpi \).

Theorem C

Let \(X',X'' \in { lT}^n_m\) be two complementary pairs of trees. Then
$$\begin{aligned} \varpi (X') = \varpi (X'') \ \left( \hbox {equality in }\mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) \right) \end{aligned}$$
if and only if
$$\begin{aligned} \pi (X') = \pi (X'') \ \left( \hbox {equality in }{ zT}^n_m\right) , \end{aligned}$$
so the image of \(\varpi : { lT}^n_m \rightarrow \mathsf{F}(\Xi )\) is isomorphic to \({ zT}^n_m\).

Our proof of this theorem occupies the rest of this section.

4.4 Proof that \(\pi (X') = \pi (X'')\) implies \(\varpi (X') = \varpi (X'')\).

Parallel to \(r_{\overline{U}, \overline{D}}(U,D,\ell ) \in { lT}^{\overline{n}}_{\overline{m}}\) of (15) there is a similar restriction \(s_{\overline{U}, \overline{D}}(U,D,z) \in { zT}^{\overline{n}}_{\overline{m}}\) defined for each \((U,D,z) \in { zT}^n_m\) and subtrees \(\overline{U} \subset U\) and \(\overline{D} \subset D\). They commute with the canonical projection in the sense that, for \(X = (U,D,\ell ) \in { lT}^n_m\),
$$\begin{aligned} \pi \left( r_{\overline{U}, \overline{D}}(X)\right) = s_{\overline{U}, \overline{D}}\left( \pi (X)\right) . \end{aligned}$$
(21)
The restriction \(s_{\overline{U},\overline{D}}\) can be defined along similar lines as \(r_{\overline{U},\overline{D}}\). The only subtlety is that the zone function restricted to \({ Vert}(\overline{U}) \cup { Vert}(\overline{D})\) need not satisfy (ii) of Definition A, so we need to identify, in its image, adjacent zones of the same type. We leave the details to the reader.

The construction of \(\varpi (X)\) given in Sect. 4.3 was divided into three cases, determined by the relative positions of the root vertices of \(U\) and \(D\). This information is, by Proposition 4.3, retained by the induced zone function of \(\pi (X)\). Therefore the case into which \(X\) falls depends only on the projection \(\pi (X)\).

Let \(X',X'' \in { lT}^n_m\) be such \(\pi (X') = \pi (X'')\). Then \(X'\) and \(X''\) may differ only by the level functions, i.e. \(X' = (U,D,\ell ')\) and \(X''= (U,D,\ell '')\). Let us proceed by induction on the number of vertices of \(D\).

If one (hence both) of \(X'\), \(X''\) falls into Case 1 or Case 2 of our construction of \(\varpi \), then clearly \(\varpi (X'') = \varpi (X'')\) since in these cases \(\varpi \) manifestly depends only on the trees \(U\) and \(D\) not on the level function. The induced zone function of \(\pi (X') = \pi (X'')\) must be of type \((\mathrm D \mathrm U)\) in Case 1 and \((\mathrm D\mathrm B\mathrm U)\) in Case 2.

In Case 3 we observe first that the exact form of the decomposition in Fig. 11 depends only on the relative positions of the root vertex of \(D\) and the vertices of \(U\). It is therefore, by Proposition 4.3, determined by the induced zone function, so it is the same for both \(X'\) and \(X''\). Now we invoke the commutativity (21) to check that
$$\begin{aligned} \pi (X'_i)&= \pi \left( r_{U_i,c^b_1}(X')\right) = s_{U_i,c^b_1}\left( \pi (X')\right) = s_{U_i,c^b_1}\left( \pi (X'')\right) = \pi \left( r_{U_i,c^b_1}(X'')\right) \\&= \pi (X''_i) \end{aligned}$$
for each \(1 \le i \le a\). Similarly we verify that \(\pi (Y'_j) = \pi (Y''_j)\) for each \(1 \le j \le b\). By the induction assumption,
$$\begin{aligned} \varpi (X'_i) = \varpi (X''_i) \ \hbox { and } \ \varpi (Y'_j) = \varpi (Y''_j)\ \hbox { for }\ 1 \le i \le a,\ 1 \le j \le b, \end{aligned}$$
therefore
$$\begin{aligned} \varpi (X') = \frac{\varpi (Y'_1) \cdots \varpi (Y'_b)}{\varpi (X'_1)\cdots \varpi (X'_a)} = \frac{\varpi (Y''_1) \cdots \varpi (Y''_b)}{\varpi (X''_1)\cdots \varpi (X''_a)} = \varpi (X''). \end{aligned}$$
This finishes our proof of the implication \(\pi (X') = \pi (X'') \Longrightarrow \varpi (X') = \varpi (X'')\).

4.5 Proof that \(\varpi (X') = \varpi (X'')\) implies \(\pi (X') = \pi (X'')\).

Let us show that \(\pi (X)\) is uniquely determined by \(\varpi (X) \in \mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) \). As we already remarked, elements of \(\mathsf{F}(\Xi )\) are represented by directed graphs \(G\) whose vertices are corollas \(c^b_a\) with \(a\) inputs and \(b\) outputs, where \(a,b \ge 1\), \((a,b) \not = (1,1)\). Let \(e\) be an internal edge of \(G\), connecting an output of \(c^s_r\) with an input of \(c^v_u\). We say that \(e\) is special if either \(s=1\) or \(u=1\). The graph \(G\) is special if all its internal edges are special. Finally, and element of \(\mathsf{F}(\Xi )\) is special if it is represented by a special graph. We have the following simple lemma whose proof immediately follows from the definition of the fraction.

Lemma 4.4

Let \(A_1, \ldots A_l, B_1, \ldots B_k \in \mathsf{F}(\Xi )\) be as in Definition 5.3. The fraction
$$\begin{aligned} \frac{A_1 \cdots A_l}{B_1 \cdots B_k} \end{aligned}$$
is special if and only if all \(A_1, \ldots A_l, B_1, \ldots B_k\) are special and if \(k=1\) or \(l=1\).

Lemma 4.4 implies that \(\varpi (X)\) is special if and only if \(X\) falls into Case 1 or Case 2 of Sect. 4.3. It is also clear that \(X\) falls into Case 2 if and only if \(\varpi (X)\) is special and the graph representing \(\varpi (X)\) has a (unique) vertex \(c^b_a\) with \(a,b \ge 2\). Therefore \(\varpi (X)\) bears the information to which case of its construction \(X = (U,D,\ell ) \in { lT}^n_m\) falls.

Suppose that \(X\) falls to Case 1 of our definition of \(\varpi (X)\). Clearly, formula (17) uniquely determines the planar up-rooted tree \(U\) and a down-rooted tree \(D\) such that \(X = (U,D,\ell )\). The only possible zone function \(z\) for \(\pi (X) = (U,D,z)\) is of type \((\mathrm D\mathrm U)\) with
$$\begin{aligned} z\left( { Vert}(D)\right) = \{1\},\ z\left( { Vert}(U)\right) = \{2\}. \end{aligned}$$
If \(X = (U,D,\ell )\) falls into Case 2 of our construction of \(\varpi (X)\), we argue as in the previous paragraph. Formula (18) uniquely determines \(\varpi (U_1),\ldots ,\varpi (U_a)\) and \(\varpi (Y_1),\ldots ,\varpi (Y_b)\) and therefore also the trees \(U_1,\ldots ,U_a,Y_1,\ldots ,Y_b\) in the decomposition in Fig. 10. Therefore also \(U\) and \(D\) are uniquely determined, and clearly the only possible zone function \(z\) for \(\pi (X) = (U,D,z)\) is of type \((\mathrm D,\mathrm B,\mathrm U)\) with
$$\begin{aligned}&z\left( { Vert}(D_j)\right) = \{1\},\ z\left( { Vert}(U_i)\right) = \{3\},\\&z(\hbox {root vertex~of }U)=\ z(\hbox {root vertex~of }D) = \{2\}, \end{aligned}$$
\(1 \le i \le a\), \(1 \le j \le b\).
Assume that \(X\) falls into Case 3 of our construction. Let us call, only for the purposes of this proof, a directed graph a generalized tree, if it is obtained by grafting directed up-rooted trees \(S_1,,,\ldots ,S_u\) into the inputs of the directed corolla \(c^v_u\), with some \(u,v \ge 1\), \((u,v) \not = (1,1)\). So a generalized tree is a directed graph of the formwhere we keep our convention that all edges are oriented to point upwards. Since \(X\) falls into Case 3, we know that \(\varpi (X)\) is as in (20), for some \(X_i\), \(Y_j\), \(1 \le i \le a\), \(1 \le j \le b\). We moreover know that the complementary pairs \(X_i\) fall into Case 1 or Case 2 of the construction of \(\varpi (X_i)\).
Now let \(G_1,\ldots ,G_r\) be the maximal generalized trees containing the inputs of the graph representing \(\varpi (X)\), numbered from left to right. It is clear from the definition of the fraction that \(r=a\) and that \(G_i\) is, for each \(1 \le i \le a\), the graph representing \(\varpi (X_i)\). So all \(\varpi (X_1),\ldots ,\varpi (X_a)\) are determined by \(\varpi (X)\). A simple argument shows that if
$$\begin{aligned} \frac{A'_1 \cdots A'_l}{B_1 \cdots B_k} = \frac{A''_1 \cdots A''_l}{B_1 \cdots B_k} \end{aligned}$$
for some \(A'_1,\ldots , A'_l,A''_1,\ldots , A''_l \in \mathsf{F}(\Xi ) \left( \begin{array}{c} *\\ k \end{array} \right) \) and \(B_1,\ldots , B_k \in \mathsf{F}(\Xi ) \left( \begin{array}{c} l\\ * \end{array} \right) \), then \(A'_i = A''_i\) for each \(1 \le i \le l\). We therefore see that also \(\varpi (Y_1),\ldots ,\varpi (Y_b)\) are determined by \(\varpi (X)\).

Let us summarize what we have. We know each \(\varpi (X_i)\). Since the construction of \(\varpi (X_i)\) falls into Case 1 or Case 2, we know, as we have already proved, the trees \(U_1,\ldots ,U_a\) in Fig. 11, and also the relative positions of vertices of \(U_1,\ldots ,U_a\) and the root vertex of \(D\).

Now we preform a similar analysis of \(\varpi (Y_1),\ldots ,\varpi (Y_b)\) and repeat this process until we get trivial trees. It is clear that, during this process, we fully reconstruct the trees \(U,D\) in \(X = (U,D,\ell )\) and the relative positions of their vertices. By Proposition 4.3, this uniquely determines the zone function in \(\pi (X) = (U,D,z)\). This finishes our proof of the second implication.

5 The particular case \(\mathcal{K }^2_m\)

In this section we analyze in detail the poset \(\mathcal{K }^2_m\) for which the notion of complementary pairs with zones takes a particularly simple form.

5.1 Trees with a diaphragm

Let us consider the ordinal \(\left\{ (-\infty ,1) < 1 < (1,+\infty )\right\} \). A diaphragm of an up-rooted tree \(U\) is an order-preserving map
$$\begin{aligned} \zeta : { Vert}(U) \rightarrow \textstyle \left\{ (-\infty ,1), 1, (1,+\infty )\right\} \end{aligned}$$
(22)
which is strictly order-preserving at \(1\). By this we mean that, if \(\zeta (v') = \zeta (v'')= 1\) then neither \(v' < v''\) nor \(v' > v''\). We will denote \({ dT}^2_m\) the set of all pairs \((U,\zeta )\), where \(U\) is an up-rooted tree with \(m\) leaves and \(\zeta \) a diaphragm.
One may imagine a tree with a diaphragm as a planar up-rooted tree crossed by a horizontal line, i.e. a diaphragm, see the rightmost column of the table in Fig. 6 for examples. It is convenient to introduce the following subsets of \({ Vert}(U)\):
$$\begin{aligned} { Vert}_{<1}(U) := \zeta ^{-1}(-\infty ,1),\ { Vert}_{1}(U) := \zeta ^{-1}(1),\ { Vert}_{>1}(U) := \zeta ^{-1}(1,+\infty ) \end{aligned}$$
and the ‘closures’
$$\begin{aligned} { Vert}_{\le 1}(U) := { Vert}_{<1}(U) \cup { Vert}_{1}(U)\ \hbox { and } { Vert}_{\ge 1}(U) := { Vert}_{>1}(U) \cup { Vert}_{1}(U). \end{aligned}$$
A morphism \(\phi : (U',\zeta ') \rightarrow (U'',\zeta '')\) of trees with a diaphragm is a morphism \(\phi : U' \rightarrow U''\) of planar up-rooted trees which preserves the closures, i.e.
$$\begin{aligned}&\phi \left( { Vert}_1(U')\right) \subset { Vert}_1(U''),\ \phi \left( { Vert}_{< 1}(U')\right) \subset { Vert}_{\le 1}(U''), \\&\quad \phi \left( { Vert}_{> 1}(U')\right) \subset { Vert}_{\ge 1}(U''). \end{aligned}$$
We say that \((U',\zeta ') < (U'',\zeta '')\) if there exists a morphism \((U',\zeta ') \rightarrow (U'',\zeta '')\).

Proposition 5.1

The posets \(({ zT}^2_m,<)\) and \(({ dT}^2_m,<)\) are, for each \(m \ge 1\), naturally isomorphic.

Proof

For Open image in new window , denote \(L\) the value of \(z\) on the vertex of Open image in new window . Define \(\zeta \) by
$$\begin{aligned} \zeta (v) := \left\{ \begin{array}{ll} (-\infty ,0)&{}\quad \hbox { if } \ z(v) < L, \\ 1&{}\quad \hbox { if }\ z(v) = L,\ \hbox {and} \\ (1,+\infty )&{}\quad \hbox { if }\ z(v) > L. \end{array} \right. \end{aligned}$$
It is easy to see that \((U,\zeta )\) is a tree with a diaphragm, that the correspondence \((U,z) \mapsto (U,\zeta )\) is one-to-one and that it preserves the partial orders. \(\square \)
The natural projection \(\pi : { lT}^2_m \rightarrow { zT}^2_m\) can be, in terms of trees with a diaphragm, described as follows. Let Open image in new window and assume that the vertex of Open image in new window is placed at level \(L\). Then \(\pi (X) := (U,\zeta )\), with the diaphragm
$$\begin{aligned} \zeta (v) := \left\{ \begin{array}{l@{\quad }l} (-\infty ,0)&{}\quad \hbox { if } \ \ell (v) < L, \\ 1&{}\quad \hbox { if }\ \ell (v) = L,\ \hbox {and} \\ (1,+\infty )&{}\quad \hbox { if } \ \ell (v) > L. \end{array} \right. \end{aligned}$$

Example

The \(\pi \)-images of complementary pairs in \({ lT}^2_3\) are listed in the rightmost column of the table in Fig. 6.

Example

Figure 12 illustrates the projection \(P^2_4 \rightarrow K^2_4\). It shows the face poset of a square face of the \(3\)-dimensional \(P^2_4\) together with the corresponding complementary pairs in \({ lT}^2_4\) and its projection, which is in this case the poset of the interval indexed by the corresponding trees with a diaphragm.

Fig. 12

The projection of a face of \(P^2_4\) to \(K^2_4\). The faces of the interval in \(K^2_4\) (right) are indexed by trees with a diaphragm

As an exercise, we recommend describing the map \(\varpi : { zT}^2_m \rightarrow \mathsf{F}(\Xi ) \left( \begin{array}{c} 2\\ m \end{array} \right) \) in terms of trees with a diaphragm. One may generalize the above description of the poset \({ zT}^n_m\) also to \(n> 2\). In this case, the tree \(U\) corresponding to \((U,D,z) \in { zT}^n_m\) may have several diaphragms, depending on the relative positions of the vertices of \(D\). The combinatorics of this kind of description becomes, however, unmanageably complicated with growing \(n\).

5.2 Relation to the multiplihedron

Multiplihedra appeared in the study of homotopy multiplicative maps between \(A_\infty \)-spaces [2, 12]. The \(m\)-th multiplihedron \(J_m\) is a convex polytope of dimension \(m-1\) whose vertices correspond to ways of bracketings \(m\) variables and applying an operation, cf. [1]. As also explained in [1], the faces of \(J_m\) are indexed by painted \(m\)-trees which are, by definition, directed (rooted) planar trees with \(m\) leaves, two types of edges—black and white—and vertices of the following two types:
  1. (i)

    vertices with at least two inputs whose all adjacent edges are of the same color, or

     
  2. (ii)

    vertices whose all inputs are white and whose output is black.

     
The set \({ pT}_m\) of all painted \(m\)-trees has a partial order \(<\) induced by contracting the edges. The poset \(\mathcal{J }_m := ({ pT}_m,<)\) is then the poset of faces of the \(m\)-th multiplihedron \(J_m\). We believe that Fig. 13 makes the above definitions clear.
Fig. 13

The faces of the multiplihedron \(J_3\) indexed by the set \({ pT}_3\) of painted \(3\)-trees. The labels of vertices in terms of bracketings of \(3\) variables and an operation \(f\) are also shown

Proposition D

The face poset \(\mathcal{K }^2_m\) of the biassociahedron \(K^2_m\) is isomorphic to the face poset \(\mathcal{J }_m\) of the multiplihedron \(J_m\), for each \(m \ge 2\).

Proof

By Proposition 5.1, it suffices to prove that the posets \(({ dT}_m,<)\) and \(({ pT}_m,<)\) are isomorphic. It is very simple. Having a tree \(U\) with a diaphragm, we paint everything that lies above14 the diaphragm black, and everything below white. If the diaphragm intersects an edge of \(U\), we introduce at the intersection a new vertex of type (ii) with one input edge. The result will obviously be a painted tree belonging to \({ pT}_m\). The isomorphism we have thus described clearly preserves the partial orders. \(\square \)

The correspondence of Proposition D is, for \(m=3\), illustrated by the two rightmost columns of the table in Fig. 6.

Remark Forcey in [1] constructed an explicit realization of the poset \(\mathcal{J }_m\) by the face poset of a convex polyhedron. This, combined with Proposition D proves, independently of [10], that \(\mathcal{K }^2_m\) is the face poset of a convex polyhedron, too.

5.3 Relation to \(K_{m-1,n-1}\)

In Sect. 3.4 we recalled that the authors of [10] labelled the vertices of the bipermutahedron by strings of pairwise block transverse matrices and in (9) we identified the strings corresponding to the two bottom vertices \(L\) and \(R\) of the square of \(P^2_4\) in Fig. 12 asSaneblidze–Umble’s \(K_{m-1,n-1}\) is the quotient of the bipermutahedron \(P^n_m\) (which they denote by \(P_{m-1,n-1}\)) by the relation given by stacking the adjacent elementary matrices of the same type. In our particular case, both strings labelling the vertices \(L\) and \(R\) are of the form \(A_1BA_2A_3\), where \(A_i\)’s are Open image in new window -matrices and \(B\) is a Open image in new window -matrix. By stacking the adjacent pair \(A_2A_3\) one replaces both Open image in new window and Open image in new window by the (non-elementary) Open image in new window -matrix Open image in new window . In this manner, the vertices \(L\) and \(R\) of \(P^2_4\) labelled by the strings in (23) are identified, representing one vertex of \(K_{2,4}\).

The relation of ‘stacking adjacent matrices’ obviously corresponds to our passage from complementary pairs of trees with levels to complementary pairs of trees with zones. This explains the relation between our step-one associahedron \(K^n_m\) and Saneblidze–Umble’s \(K_{m-1,n-1}\).

Footnotes

  1. 1.

    Other possible names are \(B_\infty \)-algebras or strongly homotopy bialgebras.

  2. 2.

    PROPs generalize operads. We briefly recall them in the Appendix.

  3. 3.

    Sometimes also spelled permutohedron.

  4. 4.

    In [10] it was denoted \(K_{m,n}\).

  5. 5.

    Denoted \(P_{m-1,n-1}\) in [10].

  6. 6.

    Strictly order-preserving means that \(v' < v''\) implies \(\ell (v') < \ell (v'')\).

  7. 7.

    By a map of trees we understand a sequence of contractions of internal edges. In particular, the root and leaves are fixed.

  8. 8.

    We however recall the correspondence between trees with levels and ordered partitions in Sect. 3.3.

  9. 9.

    Alternatively, replace the labels by balls and change the direction of gravity.

  10. 10.

    Therefore \(h = m+n-2\).

  11. 11.

    The one such that \(T' < T''\) if and only if there exists a morphism of planar up-rooted trees \(T' \rightarrow T''\).

  12. 12.

    By this we mean that neither \(u' < u''\) nor \(u' > u''\).

  13. 13.

    That is the vertex adjacent to the root.

  14. 14.

    We keep our convention that all edges are oriented to point upwards.

Notes

Acknowledgments

I would like to express my gratitude to Samson Saneblidze, Jim Stasheff, Ron Umble and the anonymous referee for reading the manuscript and offering helpful remarks and suggestions. I also enjoyed the wonderful atmosphere in the Max-Planck Institut für Matematik in Bonn during the period when the first draft of this paper was completed.

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2013

Authors and Affiliations

  1. 1.Institute of MathematicsCzech AcademyPragueThe Czech Republic
  2. 2.MFF UKPragueThe Czech Republic

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