1. Preliminaries

1.3. Incidence algebras

Let X be a locally finite poset and K a field. The incidence algebra [Reference Rota20] I(X, K) of X over K is the K-space of functions $f\,{:}\,X\times X\to K$ such that $f(x,y)=0$ if $x\nleq y$ . This is a unital K-algebra under the convolution product

\begin{equation*} (fg)(x,y)=\sum_{x\leq z\leq y}f(x,z)g(z,y), \end{equation*}

for any $f, g\in I(X,K)$ . Its identity element $\delta$ is given by

\begin{align*}\delta(x,y)=\begin{cases}1, & x=y,\\ \\[-7pt] 0, & x\ne y.\end{cases}\end{align*}

Throughout the rest of the paper, X will stand for a connected finite poset. Then I(X, K) admits the standard basis $\{e_{xy} \,{:}\, x\leq y\}$ , where

\begin{align*}e_{xy}(u,v)=\begin{cases}1, & (u,v)=(x,y),\\ \\[-7pt] 0, & (u,v)\ne(x,y).\end{cases}\end{align*}

We will write $e_{x}=e_{xx}$ . Denote also $B=\{e_{xy} \,{:}\, x<y\}$ . It is a well-known fact (see [Reference Spiegel and O’Donnell22, Theorem 4.2.5]) that the Jacobson radical of I(X, K) is

\begin{equation*} J(I(X,K))=\{f\in I(X,K) \,{:}\, f(x,x)=0 \text{ for all } x\in X\}={\rm{Span}}_K B. \end{equation*}

Diagonal elements of I(X, K) are those $f\in I(X,K)$ satisfying $f(x,y)=0$ for $x\neq y$ . They form a commutative subalgebra D(X, K) of I(X, K) spanned by $\{e_{x} \,{:}\, x \in X\}$ . Clearly, each $f\in I(X,K)$ can be uniquely written as $f=f_D+f_J$ with $f_D\in D(X,K)$ and $f_J\in J(I(X,K))$ .

1.4. Decomposition of $\boldsymbol\phi\boldsymbol\in \boldsymbol{Aut}^{\boldsymbol\pm}(\boldsymbol{{I}}(\boldsymbol{{X}}, \boldsymbol{{K}}))$

Now, we recall the descriptions of automorphisms and anti-automorphisms of I(X, K). Firstly, if X and Y are finite posets and $\lambda\,{:}\, X\to Y$ is an isomorphism (resp. anti-isomorphism), then $\lambda$ induces an isomorphism (resp. anti-isomorphism) $\hat\lambda\,{:}\, I(X,K)\to I(Y,K)$ defined by $\hat\lambda(e_{xy})=e_{\lambda(x)\lambda(y)}$ (resp. $\hat\lambda(e_{xy})=e_{\lambda(y)\lambda(x)}$ ), for all $x\leq y$ in X. An element $\sigma\in I(X,K)$ such that $\sigma(x,y)\neq 0$ , for all $x\leq y$ , and $\sigma(x,y)\sigma(y,z)=\sigma(x,z)$ whenever $x\leq y\leq z$ , determines an automorphism $M_{\sigma}$ of I(X, K) by $M_{\sigma}(e_{xy})=\sigma(x,y)e_{xy}$ , for all $x\leq y$ . Such automorphisms are called multiplicative. Any automorphism (anti-automorphism) of I(X, K) decomposes as

(1) \begin{align}\phi=\hat\lambda \circ \xi \circ M_{\sigma},\end{align}

where $\lambda\in {\rm{Aut}}(X)$ (resp. ${\rm{Aut}}^-(X)$ ), $\xi $ is an inner automorphism, and $M_{\sigma}$ is a multiplicative automorphism of I(X, K). For automorphisms, this was proved in [Reference Baclawski1, Theorem 5] and for anti-automorphisms in [Reference Brusamarello, Fornaroli and Santulo3, Theorem 5] (for more results on automorphisms and anti-automorphisms of incidence algebras see [Reference Brusamarello, Fornaroli and Santulo3, Reference Brusamarello, Fornaroli and Santulo4, Reference Brusamarello, Fornaroli and Santulo5, Reference Brusamarello and Lewis6, Reference Drozd and Kolesnik12, Reference Khripchenko15, Reference Spiegel21, Reference Stanley23]).

1.5. Lie automorphisms of incidence algebras

In this section, we introduce several new notations and recall some definitions and results from [Reference Fornaroli, Khrypchenko and Santulo13].

We denote by $\mathcal{C}(X)$ the set of maximal chains in X. Let $C\,{:}\, u_1<u_2<\dots<u_m$ in $\mathcal{C}(X)$ . A bijection $\theta\,{:}\,B\to B$ is increasing (resp. decreasing) on C if there exists $D\,{:}\, v_1<v_2<\dots<v_m$ in $\mathcal{C}(X)$ such that $\theta(e_{u_iu_j})=e_{v_iv_j}$ for all $1\le i<j\le m$ (resp. $\theta(e_{u_iu_j})=e_{v_{m-j+1}v_{m-i+1}}$ for all $1\le i<j\le m$ ). In this case, we write $\theta(C)=D$ . Moreover, we say that $\theta$ is monotone on maximal chains in X if, for any $C\in \mathcal{C}(X)$ , $\theta$ is increasing or decreasing on C. We denote by $\mathcal{M}(X)$ the set of bijections $B\to B$ which are monotone on maximal chains in X. It is easy to see that $\mathcal{M}(X)$ is a subgroup of the symmetric group S(B). Each $\theta\in\mathcal{M}(X)$ induces a bijection on $\mathcal{C}(X)$ which maps C to $\theta(C)$ .

Let $\theta\,{:}\,B\to B$ be a bijection and $X^2_<=\{(x,y)\in X^2\,{:}\, x<y\}$ . A map $\sigma\,{:}\,X^2_<\to K^*$ is compatible with $\theta$ if $\sigma(x,z)=\sigma(x,y)\sigma(y,z)$ whenever $\theta(e_{xz})=\theta(e_{xy})\theta(e_{yz})$ , and $\sigma(x,z)=-\sigma(x,y)\sigma(y,z)$ whenever $\theta(e_{xz})=\theta(e_{yz})\theta(e_{xy})$ .

Let $\theta\,{:}\,B\to B$ be a bijection and $\Gamma\,{:}\, u_0,u_1,\dots,u_m=u_0$ a closed walk in X. In [Reference Fornaroli, Khrypchenko and Santulo13], we introduced the following 4 functions $X\to\mathbb{N}$ :

\begin{align*}s^+_{\theta,\Gamma}(z)&=|\{i\,{:}\, u_i<u_{i+1}\text{ and }\exists w>z\text{ such that }\theta(e_{zw})=e_{u_iu_{i+1}}\}|,\\[2pt] s^-_{\theta,\Gamma}(z)&=|\{i\,{:}\, u_i>u_{i+1}\text{ and }\exists w>z\text{ such that }\theta(e_{zw})=e_{u_{i+1}u_i}\}|,\\[2pt] t^+_{\theta,\Gamma}(z)&=|\{i\,{:}\, u_i<u_{i+1}\text{ and }\exists w<z\text{ such that }\theta(e_{wz})=e_{u_iu_{i+1}}\}|,\\[2pt] t^-_{\theta,\Gamma}(z)&=|\{i\,{:}\, u_i>u_{i+1}\text{ and }\exists w<z\text{ such that }\theta(e_{wz})=e_{u_{i+1}u_i}\}|.\end{align*}

We call the bijection $\theta\,{:}\,B\to B$ admissible if

(2) \begin{align} s^+_{\theta,\Gamma}(z)-s^-_{\theta,\Gamma}(z)=t^+_{\theta,\Gamma}(z)-t^-_{\theta,\Gamma}(z), \end{align}

for any closed walk $\Gamma\,{:}\,u_0,u_1,\dots,u_m=u_0$ in X and for all $z\in X$ . In particular, if X is a tree, then any bijection $\theta\,{:}\,B\to B$ is admissible. We denote by $\mathcal{AM}(X)$ the set of those $\theta\in\mathcal{M}(X)$ which are admissible.

Let $X=\{x_1,\dots, x_n\}$ . Given $\theta\in \mathcal{AM}(X)$ , a map $\sigma\,{:}\,X_<^2\to K^*$ compatible with $\theta$ and a sequence $c=(c_1,\dots,c_n)\in K^n$ such that $\sum_{i=1}^nc_i\in K^*$ , we define in [Reference Fornaroli, Khrypchenko and Santulo13, Definition 5.17] the following elementary Lie automorphism $\tau=\tau_{\theta,\sigma,c}$ of I(X, K) where, for any $e_{xy}\in B$ ,

\begin{equation*}\tau(e_{xy})=\sigma(x,y)\theta(e_{xy}),\end{equation*}

and $\tau|_{D(X,K)}$ is determined by

\begin{equation*}\tau(e_{x_i})(x_1,x_1)=c_i,\end{equation*}

$i=1,\dots,n$ , as in Lemmas 5.8 and 5.16 from [Reference Fornaroli, Khrypchenko and Santulo13]. As in [Reference Fornaroli, Khrypchenko and Santulo13, Definition 5.15], we say that $\tau$ induces the pair $(\theta,\sigma)$ and in some situations we write $\theta=\theta_{\tau}$ .

As in [Reference Fornaroli, Khrypchenko and Santulo13], we denote by LAut(I(X, K)) the group of Lie automorphisms of I(X, K) and by $\widetilde{{\rm{LAut}}}(I(X,K))$ its subgroup of elementary Lie automorphisms. We will also use the notation $\textrm{Inn}_1(I(X,K))$ for the subgroup of inner automorphisms consisting of conjugations by $\beta\in I(X,K)$ with $\beta_D=\delta$ .

Theorem 1.1. [Reference Fornaroli, Khrypchenko and Santulo13, Theorem 4.15]The group LAut(I(X, K)) is isomorphic to the semidirect product ${\rm{Inn}}_1(I(X, K))\rtimes\widetilde{{\rm{LAut}}}(I(X, K))$ .

2. Proper Lie automorphisms of I(X, K) and proper bijections of B

Let $\varphi\in {\rm{LAut}}(I(X, K))$ . Then $\varphi=\psi\circ \tau_{\theta,\sigma,c}$ , where $\psi\in {\rm{Inn}}_1(I(X, K))$ and $\tau_{\theta,\sigma,c}$ is an elementary Lie automorphism of I(X, K), by Theorem 1.1. Note that $\varphi$ is proper if and only if $\tau_{\theta,\sigma,c}$ is proper. Therefore, all Lie automorphisms of I(X, K) are proper if and only if all elementary Lie automorphisms of I(X, K) are proper.

Let $\varphi=\tau_{\theta,\sigma,c}$ be an elementary Lie automorphism of I(X, K). Suppose that $\varphi$ is proper, $\varphi=\phi+\nu$ , where $\phi\in {\rm{Aut}}(I(X, K))$ or $-\phi\in {\rm{Aut}}^-(I(X, K))$ and $\nu$ is a linear central-valued map on I(X, K) such that $\nu([I(X, K),I(X, K)])=\{0\}$ . If $x<y$ , then $e_{xy}\in J(I(X, K))=[I(X, K),I(X, K)]$ , by [Reference Fornaroli, Khrypchenko and Santulo13, Proposition 2.3]. Thus

(3) \begin{align}\varphi(e_{xy})=\phi(e_{xy}), \forall x<y.\end{align}

By [Reference Spiegel and O’Donnell22, Corollary 1.3.15], for each $x\in X$ there is $\alpha_x\in K$ such that

(4) \begin{align}\varphi(e_{x})=\phi(e_{x})+\alpha_x\delta.\end{align}

Suppose firstly that $\phi\in {\rm{Aut}}(I(X, K))$ . Then, by (1), $\phi=\hat\lambda \circ \xi_f \circ M_{\tau}$ , where $\lambda\in {\rm{Aut}}(X)$ , $\xi_f$ is an inner automorphism and $M_{\tau}$ is a multiplicative automorphism of I(X, K). Thus, by (3),

(5) \begin{align} \theta(e_{xy})= & \sigma(x,y)^{-1}(\hat\lambda \circ \xi_f \circ M_{\tau})(e_{xy})=\sigma(x,y)^{-1}(\hat\lambda \circ \xi_f)(\tau(x,y)e_{xy}) \nonumber\\[3pt] = & \sigma(x,y)^{-1}\hat\lambda(\tau(x,y)fe_{xy}f^{-1})=\sigma(x,y)^{-1}\tau(x,y)\hat\lambda(f)e_{\lambda(x)\lambda(y)}\hat\lambda(f)^{-1}.\end{align}

Analogously, if $-\phi\in {\rm{Aut}}^-(I(X, K))$ , then, by (1), $\phi=\hat\lambda \circ \xi_f \circ M_{\tau}$ , where $\lambda\in {\rm{Aut}}^-(X)$ , $\xi_f$ is an inner automorphism and $M_{\tau}$ is a multiplicative automorphism of I(X, K). Thus, by (3),

(6) \begin{align} \theta(e_{xy})= \sigma(x,y)^{-1}(\hat\lambda \circ \xi_f \circ M_{\tau})(e_{xy})= \sigma(x,y)^{-1}\tau(x,y)\hat\lambda(f)^{-1}e_{\lambda(y)\lambda(x)}\hat\lambda(f).\end{align}

Proof. The map sending $\lambda\in {\rm{Aut}}(X)$ (resp. $\lambda\in {\rm{Aut}}^-(X)$ ) to $\theta\in\mathcal{P}(X)$ , such that $\theta(e_{xy})=e_{\lambda(x)\lambda(y)}$ (resp. $\theta(e_{xy})=e_{\lambda(y)\lambda(x)}$ ), is an epimorphism from ${\rm{Aut}}^\pm(X)$ to $\mathcal{P}(X)$ . We only need to prove that it is injective.

It is injective on Aut(X). Indeed, take $\lambda,\mu\in {\rm{Aut}}(X)$ such that $(\lambda(x),\lambda(y))=(\mu(x),\mu(y))$ for all $x<y$ in X. Let x be an arbitrary element of X. Since X is connected and $|X|>1$ , there is $y\in X$ such that either $y<x$ or $y>x$ . In both cases, we get $\lambda(x)=\mu(x)$ . Thus, $\lambda=\mu$ . Similarly, one proves injectivity on ${\rm{Aut}}^-(X)$ .

Assume now that there are $\lambda\in {\rm{Aut}}(X)$ and $\mu\in {\rm{Aut}}^-(X)$ such that $(\lambda(x),\lambda(y))=(\mu(y),\mu(x))$ for all $x<y$ in X. We first show that X must have length at most 1. Indeed, if there were $x<y<z$ in X, then we would have $(\lambda(x),\lambda(y))=(\mu(y),\mu(x))$ and $(\lambda(y),\lambda(z))=(\mu(z),\mu(y))$ , whence $\lambda(x)=\lambda(z)$ , a contradiction. Now, consider a triple $x,y,z\in X$ with $x>z<y$ . It follows from $(\lambda(z),\lambda(x))=(\mu(x),\mu(z))$ and $(\lambda(z),\lambda(y))=(\mu(y),\mu(z))$ that $\mu(x)=\mu(y)$ , a contradiction. Similarly, the existence of a triple $x,y,z\in X$ with $x<z>y$ leads to $\lambda(x)=\lambda(y)$ . If there are two incomparable elements $x,y\in X$ , then there exists a sequence $x=x_1,\dots,x_m=y$ , where $m\ge 3$ and either $x_1<x_2>x_3$ or $x_1>x_2<x_3$ . In both cases, we come to a contradiction. Thus, X is of length at most 1 and any two elements of X are comparable, which means that X is either a singleton or a chain of length 1.

Proof. Let $\theta\in\mathcal{P}(X)$ . Then there exists $\lambda\in {\rm{Aut}}^\pm(X)$ as in Definition 2.2. In both cases, $\theta$ is the restriction of $\hat{\lambda}$ to B and, since $\hat{\lambda}$ or $-\hat{\lambda}$ is an elementary Lie automorphism of I(X, K) by [Reference Fornaroli, Khrypchenko and Santulo13, Remark 4.8], then $\theta\in\mathcal{AM}(X)$ by [Reference Fornaroli, Khrypchenko and Santulo13, Remark 5.10].

Proof. Assume first that $\theta(e_{xy})=e_{\lambda(x)\lambda(y)}$ for some $\lambda\in {\rm{Aut}}(X)$ . Then $\theta$ is increasing on any maximal chain in X and thus $\sigma(x,z)=\sigma(x,y)\sigma(y,z)$ for all $x<y<z$ . Extending $\sigma$ to $X^2_\le=\{(x,y) \,{:}\, x\le y\}$ by means of $\sigma(x,x)=1$ for all $x\in X$ , we obtain the multiplicative automorphism $M_\sigma\in {\rm{Aut}}(I(X,K))$ . Consider $\psi=\varphi\circ M^{-1}_\sigma$ . Notice that $\psi(e_{xy})=e_{\lambda(x)\lambda(y)}$ for all $x<y$ and $\psi(e_x)=\varphi(e_x)$ for all $x\in X$ . It suffices to prove that $\psi$ is proper. Indeed, if $\psi=\phi+\nu$ , then $\varphi=\psi\circ M_\sigma=\phi\circ M_\sigma+\nu$ since $M_\sigma$ is identity on D(X, R).

Clearly, $\psi(e_{xy})=\hat\lambda(e_{xy})$ for all $x<y$ , so $\hat\lambda$ is a candidate for $\phi$ . It remains to prove (4) for $\psi$ , that is, to show that $\psi(e_x)=e_{\lambda(x)}+\alpha_x\delta$ for some $\alpha_x\in K$ . The latter is equivalent to

(7) \begin{align} \psi(e_x)(y,y)=\psi(e_x)(\lambda(x),\lambda(x))-1, \end{align}

for all $y\ne \lambda(x)$ . Given $x,y\in X$ with $y\ne \lambda(x)$ , we choose a path $\lambda(x)=u_0,\dots,u_m=y$ from $\lambda(x)$ to y. Let $v_i\in X$ such that $\lambda(v_i)=u_i$ , $0\le i\le m$ . In particular, $v_0=x$ . Then

(8) \begin{align} \nonumber\\[-35pt] \psi(e_x)(y,y)=\psi(e_x)(\lambda(x),\lambda(x))+\sum_{i=0}^{m-1}(\psi(e_x)(u_{i+1},u_{i+1})-\psi(e_x)(u_i,u_i)). \end{align}

If $u_0<u_1$ , then $\theta(e_{xv_1})=e_{u_0u_1}$ , so $\psi(e_x)(u_1,u_1)-\psi(e_x)(u_0,u_0)=-1$ by [Reference Fornaroli, Khrypchenko and Santulo13, Lemma 5.7]. Similarly, if $u_0>u_1$ , then $\theta(e_{v_1x})=e_{u_1u_0}$ , so $\psi(e_x)(u_0,u_0)-\psi(e_x)(u_1,u_1)=1$ . In any case $\psi(e_x)(u_1,u_1)-\psi(e_x)(u_0,u_0)=-1$ . Observe that $v_i\ne x$ for all $i>0$ . Hence $\psi(e_x)(u_{i+1},u_{i+1})-\psi(e_x)(u_i,u_i)=0$ for all such i by [Reference Fornaroli, Khrypchenko and Santulo13, Lemma 5.7]. It follows that the sum on the right-hand side of (8) has only one non-zero term which equals $-1$ , proving (7).

The case $\theta(e_{xy})=e_{\lambda(y)\lambda(x)}$ , where $\lambda\in {\rm{Aut}}^-(X)$ , is similar.

Conversely, suppose that $\varphi=\phi+\nu$ , where $\phi\in {\rm{Aut}}(I(X,K))$ (resp. $-\phi\in {\rm{Aut}}^-(I(X,K))$ ) and $\nu$ is a linear central-valued map on I(X, K) annihilating [I(X, K), I(X, K)]. It follows from (5) (resp. (6)) and Remark 2.1 that $\theta(e_{xy})=e_{\lambda(x)\lambda(y)}$ ( $\theta(e_{xy})=e_{\lambda(y)\lambda(x)}$ ) for all $e_{xy}\in B$ , where $\lambda\in {\rm{Aut}}(X)$ ( $\lambda\in {\rm{Aut}}^-(X)$ ). Therefore, $\theta\in\mathcal{P}(X)$ .

Proof. We first notice that there is $z\in C_1$ such that $z<x$ ( $y<z$ ) if, and only if, there is $z'\in C_2$ such that $z'<x$ ( $y<z'$ ), by the maximality of $C_1$ and $C_2$ . Suppose that are $z\in C_1$ and $z'\in C_2$ such that $z,z'<x$ . Then $\theta(e_{zx})\theta(e_{xy})=\theta(e_{zy})$ and $\theta(e_{xy})\theta(e_{z\prime x})=\theta(e_{z^{\prime}y})$ . Thus, there are $s<u<v<t$ such that $\theta(e_{zx})=e_{su}$ , $\theta(e_{xy})=e_{uv}$ , $\theta(e_{z^{\prime}x})=e_{vt}$ and $\theta(e_{zy})=e_{sv}$ , $\theta(e_{z^{\prime}y})=e_{ut}$ . If $\theta^{-1}$ is increasing on a maximal chain containing $s<u<v<t$ , then $\theta^{-1}(e_{st})=\theta^{-1}(e_{su})\theta^{-1}(e_{ut})=e_{zx}e_{z^{\prime}y}$ which implies $z'=x$ , a contradiction. If $\theta^{-1}$ is decreasing on a maximal chain containing $s<u<v<t$ , then $\theta^{-1}(e_{st})=\theta^{-1}(e_{vt})\theta^{-1}(e_{sv})=e_{z^{\prime}x}e_{zy}$ which implies $z=x$ , a contradiction. Therefore, x is the minimum of $C_1$ and $C_2$ . Analogously, y is the maximum of $C_1$ and $C_2$ .

Proof. Let $\theta\in\mathcal{M}(X)$ and $\mathcal C_i$ ( $\mathcal C_d$ ) be the set of all maximal chains in X on which $\theta$ is increasing (decreasing). Let $(x,y)\in X^2_<$ . If $x\in {\rm{Min}}(X)$ , we set $\sigma(x,y)=1$ . Otherwise, by Lemma 2.7, both x and y belong only to maximal chains from $\mathcal{C}_i$ or only to maximal chains from $\mathcal{C}_d$ . In the former case we set $\sigma(x,y)=1$ and, in the latter one, we set $\sigma(x,y)=-1$ .

Let $x<y<z$ in X. Again, by Lemma 2.7, those three elements can be simultaneously only in maximal chains from $\mathcal{C}_i$ or only in maximal chains from $\mathcal{C}_d$ . If they belong to maximal chains from $\mathcal{C}_i$ , then $\sigma(x,z)=1=\sigma(x,y)\sigma(y,z)$ . Otherwise, there are two situations to be considered: $x\in {\rm{Min}}(X)$ or $x\not\in {\rm{Min}}(X)$ . In the former case, $\sigma(x,y)\sigma(y,z)=1\cdot(-1)=-1=-\sigma(x,z)$ . In the latter case, $\sigma(x,y)\sigma(y,z)=(-1)^2=1=-\sigma(x,z)$ . Thus, $\sigma$ is compatible with $\theta$ .

Proof. If $\varphi\in\widetilde{{\rm{LAut}}}(I(X,K))$ , then $\theta_\varphi\in\mathcal{AM}(X)$ by Lemma 5.4 and Remark 5.10 from [Reference Fornaroli, Khrypchenko and Santulo13]. Let now $\theta\in\mathcal{AM}(X)$ . By Lemma 2.8, there is $\sigma\,{:}\,X^2_<\to K^*$ compatible with $\theta$ . Choose an arbitrary $c=(c_1,\dots,c_n)\in K^n$ with $\sum_{i=1}^n c_i\in K^*$ , where $n=|X|$ . Then $\tau=\tau_{\theta,\sigma,c}\in \widetilde{{\rm{LAut}}}(I(X,K))$ such that $\theta_\tau=\theta$ .

Proof. Suppose that every Lie automorphism of I(X, K) is proper. Let $\theta\in\mathcal{AM}(X)$ . By Lemma 2.8, there is $\sigma\,{:}\,X^2_<\to K^*$ compatible with $\theta$ and, by [Reference Fornaroli, Khrypchenko and Santulo13, Lemma 5.16], there is $\varphi\in \widetilde{{\rm{LAut}}}(I(X,K))$ inducing $(\theta,\sigma)$ . By hypothesis, $\varphi$ is proper. Therefore, $\theta\in\mathcal{P}(X)$ by Lemma 2.6.

Conversely, suppose that $\mathcal{P}(X)=\mathcal{AM}(X)$ . Let $\varphi\in \widetilde{{\rm{LAut}}}(I(X,K))$ inducing the pair $(\theta,\sigma)$ . By hypothesis, $\theta$ is proper. Therefore, $\varphi$ is proper, by Lemma 2.6. Thus, every Lie automorphism of I(X, K) is proper.

3. Admissible bijections of B and maximal chains in X

Observe that the definitions of the functions $s^{\pm}_{\theta,\Gamma}$ and $t^{\pm}_{\theta,\Gamma}$ make sense for any sequence $\Gamma\,{:}\,u_0,\dots,u_m$ such that either $u_i<u_{i+1}$ or $u_{i+1}<u_i$ for all $0\le i\le m-1$ . We will call such sequences $\Gamma$ semiwalks. If moreover $u_0=u_m$ , then $\Gamma$ will be called a closed semiwalk.

Lemma 3.1. Let $\theta\in\mathcal{M}(X)$ , $z\in X$ and $\Gamma\,{:}\, u_0,u_1,\dots,u_m=u_0$ a closed semiwalk in X. Let $0{\le}k<k{+}l{\le}m$ such that $u_k<u_{k+1}<\dots<u_{k+l}$ or $u_k>u_{k+1}>\dots>u_{k+l}$ and set $\Gamma'\,{:}\,u_0,\dots,u_k,u_{k+l},\dots,u_m=u_0$ . Then

(9) \begin{align} \nonumber\\[-16pt] s^+_{\theta,\Gamma}(z)-t^+_{\theta,\Gamma}(z)=s^+_{\theta,\Gamma'}(z)-t^+_{\theta,\Gamma'}(z),\ s^-_{\theta,\Gamma}(z)-t^-_{\theta,\Gamma}(z)=s^-_{\theta,\Gamma'}(z)-t^-_{\theta,\Gamma'}(z). \end{align}

Proof. Assume that $u_k<u_{k+1}<\dots<u_{k+l}$ . There are two cases.

Case 1. $\theta^{-1}$ is increasing on a maximal chain containing $u_k<u_{k+1}<\dots<u_{k+l}$ . Then there are $v_k<v_{k+1}<\dots<v_{k+l}$ such that $\theta^{-1}(e_{u_iu_j})=e_{v_iv_j}$ for all $k\le i<j\le k+l$ .

Case 1.1. $z=v_i$ for some $k<i<k+l$ . If $\theta(e_{zw})=e_{u_ju_{j+1}}$ for $w>z$ and $k\le j<k+l$ , then $(z,w)=(v_j,v_{j+1})$ , which implies that $j=i$ and $w=v_{i+1}$ . Similarly $\theta(e_{wz})=e_{u_ju_{j+1}}$ for $w<z$ and $k\le j<k+l$ yields $w=v_{i-1}$ . Since, moreover, $\theta(e_{v_kv_{k+l}})=e_{u_ku_{k+l}}$ and $z\ne v_k$ , there is no $w>z$ such that $\theta(e_{zw})=e_{u_ku_{k+l}}$ . Similarly, there is no $w<z$ such that $\theta(e_{wz})=e_{u_ku_{k+l}}$ . Therefore, $s^+_{\theta,\Gamma'}(z)=s^+_{\theta,\Gamma}(z)-1$ , $t^+_{\theta,\Gamma'}(z)=t^+_{\theta,\Gamma}(z)-1$ , $s^-_{\theta,\Gamma'}(z)=s^-_{\theta,\Gamma}(z)$ and $t^-_{\theta,\Gamma'}(z)=t^-_{\theta,\Gamma}(z)$ .

Case 1.2. $z=v_k$ . Again, if $\theta(e_{zw})=e_{u_ju_{j+1}}$ for $w>z$ and $k\le j<k+l$ , then $w=v_{k+1}$ . However, there is no $w<z$ such that $\theta(e_{wz})=e_{u_ju_{j+1}}$ for some $k\le j<k+l$ , but there is a unique $w>z$ (namely, $w=v_{k+l}$ ) such that $\theta(e_{zw})=e_{u_ku_{k+l}}$ . This means that $s^\pm_{\theta,\Gamma'}(z)=s^\pm_{\theta,\Gamma}(z)$ and $t^\pm_{\theta,\Gamma'}(z)=t^\pm_{\theta,\Gamma}(z)$ .

Case 1.3. $z=v_{k+l}$ . This case is similar to Case 1.2. We have $s^\pm_{\theta,\Gamma'}(z)=s^\pm_{\theta,\Gamma}(z)$ and $t^\pm_{\theta,\Gamma'}(z)=t^\pm_{\theta,\Gamma}(z)$ .

Case 1.4. $z\not\in\{v_k,\dots,v_{k+l}\}$ . Then there is neither $w<z$ such that $\theta(e_{wz})=e_{u_ju_{j+1}}$ nor $w>z$ such that $\theta(e_{zw})=e_{u_ju_{j+1}}$ for some $k\le j<k+l$ . Moreover, there is neither $w<z$ such that $\theta(e_{wz})=e_{u_ku_{k+l}}$ nor $w>z$ such that $\theta(e_{zw})=e_{u_ku_{k+l}}$ . Thus, $s^\pm_{\theta,\Gamma'}(z)=s^\pm_{\theta,\Gamma}(z)$ and $t^\pm_{\theta,\Gamma'}(z)=t^\pm_{\theta,\Gamma}(z)$ .

Case 2. $\theta^{-1}$ is decreasing on a maximal chain containing $u_k<u_{k+1}<\dots<u_{k+l}$ . Then everything from Case 1 remains valid with the replacement of the “ $+$ ”-functions by their “minus;”-analogs and vice versa.

In any case, $s^+_{\theta,\Gamma}(z)-t^+_{\theta,\Gamma}(z)$ and $s^-_{\theta,\Gamma}(z)-t^-_{\theta,\Gamma}(z)$ are invariant under the change of $\Gamma$ for $\Gamma'$ . When $u_k>u_{k+1}>\dots>u_{k+l}$ , the proof is analogous.

Proof. The “if” part is trivial. Let us prove the “only if” part. Indeed, the case $m=2$ is explained in the proof of [Reference Fornaroli, Khrypchenko and Santulo13, Lemma 5.13], and if $m\ge 3$ , then $\Gamma$ can be extended to a closed walk $\Delta$ by inserting increasing (if $u_i<u_{i+1}$ ) or decreasing (if $u_i>u_{i+1}$ ) sequences of elements between $u_i$ and $u_{i+1}$ for all $0\le i\le m-1$ . Since (2) holds for $\Delta$ , then by Lemma 3.1 it holds for $\Gamma$ too.

Proof. We will prove (i), the proof of (ii) is analogous. Assume that $x_i\ne x_j$ for all $j\ne i$ . If $u_i<u_{i+1}$ , then $s^+_{\theta^{-1},\Gamma}(x_i)=1$ and $s^-_{\theta^{-1},\Gamma}(x_i)=0$ , since $\theta^{-1}(e_{x_iy_i})=e_{u_iu_{i+1}}$ and $\theta^{-1}(e_{x_iw})\ne e_{u_ju_{j+1}}$ for any $w\ne y_i$ and $j\ne i$ (otherwise $x_i$ would coincide with some $x_j$ for $j\ne i$ ). Similarly, if $u_i>u_{i+1}$ , then $s^+_{\theta^{-1},\Gamma}(x_i)=0$ and $s^-_{\theta^{-1},\Gamma}(x_i)=1$ . Obviously, $t^{\pm}_{\theta^{-1},\Gamma}(x_i)=0$ , because $x_i\in {\rm{Min}}(X)$ . Thus, (2) fails for the triple $(\theta^{-1},\Gamma,x_i)$ , a contradiction.

Proof. Let $C\,{:}\,x_1<\dots<x_n$ , $D\,{:}\,y_1<\dots<y_m$ and $x=x_i=y_j$ for some $1\le i\le n$ and $1\le j\le m$ . Suppose that $1<i<n$ . Then $1<j<m$ , since otherwise D would not be maximal. There exist maximal chains $C'\,{:}\,u_1<\dots<u_n$ and $D'\,{:}\,v_1<\dots<v_m$ such that $\theta(e_{x_kx_l})=e_{u_ku_l}$ for all $1\le k<l\le n$ and $\theta(e_{y_py_q})=e_{v_{m-q+1}v_{m-p+1}}$ for all $1\le p<q\le m$ . In particular, $\theta(e_{x_{i-1}x_i})=e_{u_{i-1}u_i}$ , $\theta(e_{x_ix_{i+1}})=e_{u_iu_{i+1}}$ , $\theta(e_{y_{j-1}y_j})=e_{v_{m-j+1}v_{m-j+2}}$ , $\theta(e_{y_jy_{j+1}})=e_{v_{m-j}v_{m-j+1}}$ . Observe that $x_{i-1}<x_i=y_j<y_{j+1}$ . Then either $u_i=v_{m-j}$ , or $u_{i-1}=v_{m-j+1}$ , depending on whether $\theta$ is increasing or decreasing on a maximal chain containing $x_{i-1}<x_i<y_{j+1}$ . Similarly, considering $y_{j-1}<y_j=x_i<x_{i+1}$ we obtain $v_{m-j+1}=u_{i+1}$ or $v_{m-j+2}=u_i$ . If $u_i=v_{m-j}$ , then $v_{m-j+1}=u_{i+1}$ , so that $\{u_i,u_{i+1}\}\subseteq C'\cap D'$ . However, $\theta^{-1}$ is increasing on C $^{\prime}$ and decreasing on D $^{\prime}$ , so $u_i$ is the common minimum of C $^{\prime}$ and D $^{\prime}$ and $u_{i+1}$ is the common maximum of C $^{\prime}$ and D $^{\prime}$ by Lemma 2.7. This contradicts the assumption $1<i<n$ . Similarly, $u_{i-1}=v_{m-j+1}$ implies $v_{m-j+2}=u_i$ , whence $\{u_{i-1},u_i\}\subseteq C'\cap D'$ leading to a contradiction.

Thus, $i\in\{1,n\}$ . If $i=1$ , then necessarily $j=1$ , as otherwise C would not be maximal. Similarly, if $i=n$ , then $j=m$ .

Proof. Let $\theta$ be increasing on C. Then it is increasing on D by Lemma 3.4. Using the same idea as in the proof of Lemma 3.4, we have $\theta(e_{x_{i-1}x_i})=e_{u_{i-1}u_i}$ , $\theta(e_{x_ix_{i+1}})=e_{u_iu_{i+1}}$ , $\theta(e_{y_{j-1}y_j})=e_{v_{j-1}v_j}$ , $\theta(e_{y_jy_{j+1}})=e_{v_jv_{j+1}}$ . Considering $x_{i-1}<x_i=y_j<y_{j+1}$ we conclude that $u_i=v_j$ or $u_{i-1}=v_{j+1}$ . Similarly it follows from $y_{j-1}<y_j=x_i<x_{i+1}$ that $u_i=v_j$ or $v_{j-1}=u_{i+1}$ . If $u_i\ne v_j$ , then $u_{i-1}=v_{j+1}$ and $v_{j-1}=u_{i+1}$ . But this is impossible, since $u_{i-1}<u_{i+1}$ and $v_{j+1}>v_{j-1}$ .

The proof for the decreasing case is analogous.

Proof. Let $C,D\in\mathcal{C}(X)$ be linked. Then $\theta(C)$ and $\theta(D)$ are linked by Lemma 3.5. It follows that $C\sim D$ implies $\theta(C)\sim\theta(D)$ , which induces a map $\widetilde\theta\,{:}\,\mathcal{C}(X)/{\sim}\to\mathcal{C}(X)/{\sim}$ . It is a bijection whose inverseis $\widetilde{\theta^{-1}}$ .

Assume that $\theta$ is increasing on $C\in\mathcal{C}(X)$ . Then by Lemma 3.5 it is increasing on any $D\in\mathcal{C}(X)$ which is linked to C. By the obvious induction, this extends to any $D\sim C$ . The decreasing caseis similar.

Indeed, assume that $x\in {\rm{supp}}(\mathfrak{C})\cap {\rm{supp}}(\mathfrak{D})$ , where $x\not\in {\rm{Min}}(X)$ and $x\not\in {\rm{Max}}(X)$ . There are $C\in\mathfrak{C}$ and $D\in\mathfrak{D}$ such that $x\in C\cap D$ . But then C and D are linked, so $C\sim D$ , whence $\mathfrak{C}=\mathfrak{D}$ .

Theorem 3.10. Let $\theta\in\mathcal{AM}(X)$ and $\mathfrak{C}\in\mathcal{C}(X)/{\sim}$ . Then there exists an isomorphism or an anti-isomorphism of posets $\lambda\,{:}\,{\rm{supp}}(\mathfrak{C})\to {\rm{supp}}(\widetilde\theta(\mathfrak{C}))$ such that for all $x<y$ from ${\rm{supp}}(\mathfrak{C})$ one has

(10) \begin{align} \theta(e_{xy})=\hat\lambda(e_{xy}). \end{align}

Proof. In view of Lemma 3.7, we may assume that $\theta$ is increasing on all $C\in\mathfrak{C}$ or decreasing on all $C\in\mathfrak{C}$ . Consider the case of an increasing $\theta$ . We are going to construct the corresponding $\lambda\,{:}\,{\rm{supp}}(\mathfrak{C})\to {\rm{supp}}(\widetilde\theta(\mathfrak{C}))$ . Let $x\in {\rm{supp}}(\mathfrak{C})$ and $C\,{:}\,x_1<\dots<x_n$ a maximal chain from $\mathfrak{C}$ containing x. Denote by $C'\,{:}\,u_1<\dots<u_n$ the image of C under $\theta$ . If $x=x_i$ for some $1\le i\le n$ , then we put $\lambda(x_i)=u_i$ . We still need to show that the definition does not depend on the choice of C.

If $1<i<n$ , then this is true by Lemma 3.5.

If $i=1$ , then $x\in {\rm{Min}}(X)$ . If there exists another maximal chain $D\,{:}\,y_1<\dots<y_m$ from $\mathfrak{C}$ containing x, then $x=y_1$ . We thus need to show that $u_1=v_1$ , where $D'\,{:}\,v_1<\dots<v_m$ is the image of D under $\theta$ . Since $C\sim D$ , there are $C=C_1,\dots,C_k=D$ such that $C_j$ and $C_{j+1}$ are linked for all $1\le j\le k-1$ . Denote by $z_j$ an element of $C_j\cap C_{j+1}$ , $1\le j\le k-1$ , which is neither minimal nor maximal in X. Set also $z_0=z_k=x$ . Observe that $z_j,z_{j+1}\in C_{j+1}$ for all $0\le j\le k-1$ , so that either $z_j\le z_{j+1}$ or $z_j\ge z_{j+1}$ . Let $\Gamma\,{:}\,z_0,z_1,\dots,z_k=z_0$ . Clearly, $z_0\ne z_j$ and $z_j\ne z_k$ for all $1\le j\le k-1$ , as $z_0=z_k\in {\rm{Min}}(X)$ , while $z_j\not\in {\rm{Min}}(X)$ . Moreover, we will assume that $z_j\ne z_{j+1}$ for all $1\le j\le k-2$ , since otherwise we may just remove the repetitions (and at least 2 elements will remain). Let also $a_j<b_j$ such that $\theta(e_{z_jz_{j+1}})=e_{a_jb_j}$ if $z_j<z_{j+1}$ , and $\theta(e_{z_{j+1}z_j})=e_{a_jb_j}$ if $z_j>z_{j+1}$ , $0\le j\le k-1$ . Observe that $a_0=u_1$ , since $x=z_0<z_1\in C$ , and $a_{k-1}=v_1$ , since $x=z_k<z_{k-1}\in D$ . In particular, $a_0,a_{k-1}\in {\rm{Min}}(X)$ . Since $z_j$ is not minimal for all $1\le j\le k-2$ , then neither is $a_j$ , so that $a_j\not\in\{a_0,a_{k-1}\}$ for such j. But then we must have $a_0=a_{k-1}$ , that is, $u_1=v_1$ , by Lemma 3.3 (i).

The case $i=n$ is similar. The map $\lambda\,{:}\,{\rm{supp}}(\mathfrak{C})\to {\rm{supp}}(\widetilde\theta(\mathfrak{C}))$ is thus constructed.

We now prove that $\lambda(x)<\lambda(y)$ and (10) holds for all $x<y$ from ${\rm{supp}}(\mathfrak{C})$ . By construction, this is true for x and y belonging to the same $C\in\mathfrak{C}$ . Let now $x<y$ be arbitrary elements of ${\rm{supp}}(\mathfrak{C})$ . Choose $C\in\mathfrak{C}$ containing x and $D\in\mathfrak{C}$ containing y. If $x\not\in {\rm{Min}}(X)$ , then any $C'\in\mathcal{C}(X)$ containing x and y is linked to C, so that $C'\in\mathfrak{C}$ . The case when $y\not\in {\rm{Max}}(X)$ is similar. Let now $x\in {\rm{Min}}(X)$ and $y\in {\rm{Max}}(X)$ . As above, we choose $C=C_1,\dots,C_k=D$ , such that $C_j,C_{j+1}\in\mathfrak{C}$ are linked for all $1\le j\le k-1$ , and $z_j\in C_j\cap C_{j+1}$ , $1\le j\le k-1$ , which is neither minimal nor maximal in X. We set $z_0=x$ , $z_k=y$ and $\Gamma\,{:}\,z_0,z_1,\dots,z_k,z_{k+1}=z_0$ . We also assume that $z_j\ne z_{j+1}$ for all $0\le j\le k$ and denote by $a_j<b_j$ the elements satisfying $\theta(e_{z_jz_{j+1}})=e_{a_jb_j}$ if $z_j<z_{j+1}$ and $\theta(e_{z_{j+1}z_j})=e_{a_jb_j}$ if $z_j>z_{j+1}$ , $0\le j\le k$ . As above, observe that $a_0,a_k\in {\rm{Min}}(X)$ , while $a_1,\dots,a_{k-1}\not\in {\rm{Min}}(X)$ . Then $a_0=a_k$ by Lemma 3.3 (i). Similarly it follows from Lemma 3.3 (ii) that $b_{k-1}=b_k$ . But $a_0=\lambda(x)$ and $b_{k-1}=\lambda(y)$ , since $e_{a_0b_0}=\theta(e_{z_0z_1})=e_{\lambda(z_0)\lambda(z_1)}=e_{\lambda(x)\lambda(z_1)}$ and $e_{a_{k-1}b_{k-1}}=\theta(e_{z_{k-1}z_k})=e_{\lambda(z_{k-1})\lambda(z_k)}=e_{\lambda(z_{k-1})\lambda(y)}$ . Hence, $\lambda(x)=a_k<b_k=\lambda(y)$ and $\theta(e_{xy})=\theta(e_{z_{k+1}z_k})=e_{a_kb_k}=e_{\lambda(x)\lambda(y)}=\hat\lambda(e_{xy})$ .

It is clear that $\lambda$ is a bijection whose inverse is the map $\mu\,{:}\,{\rm{supp}}(\widetilde\theta(\mathfrak{C}))\to {\rm{supp}}(\mathfrak{C})$ corresponding to $\theta^{-1}$ . Thus, $\lambda$ is an isomorphism between ${\rm{supp}}(\mathfrak{C})$ and ${\rm{supp}}(\widetilde\theta(\mathfrak{C}))$ .

The case of a decreasing $\theta$ is analogous.

The following example shows that the admissibility of $\theta$ in Theorem 3.10 cannot be dropped.

Then $\mathcal{C}(X)/{\sim}$ consists of 2 classes whose supports are $Y=\{1,\dots,10\}$ and $Z=\{1',\dots,9',10,7''\}$ . Observe that there exists $\theta\in\mathcal{M}(X)$ mapping one ${\sim}$ -class to another. It is defined as follows: $\theta(e_{ij})=e_{i'j'}$ for all $i\le j$ in X with $(i,j)\ne (5,7)$ , $\theta(e_{i'j'})=e_{ij}$ for all $i'\le j'$ in X with $(i',j')\ne(5',7'')$ , $\theta(e_{57})=e_{5'7''}$ and $\theta(e_{5'7''})=e_{57}$ (to make the definition shorter, we set $10':=10$ ). However, Y and Z are not isomorphic or anti-isomorphic because $|Y|\ne|Z|$ . The reason is that $\theta\not\in\mathcal{AM}(X)$ . Indeed, for $\Gamma\,{:}\,5<7>6<8>5$ we have $s^\pm_{\theta,\Gamma}(7')=0$ , $t^+_{\theta,\Gamma}(7')=0$ and $t^-_{\theta,\Gamma}(7')=1$ .

As a consequence of Theorems 2.10 and 3.10, we have the following result which generalizes [Reference Fornaroli, Khrypchenko and Santulo13, Corollary 5.19], where X was a chain.

Observe, however, that the condition $|\mathcal{C}(X)/{\sim}|=1$ is not necessary for all $\varphi\in {\rm{LAut}}(I(X,K))$ to be proper, as the following example shows.

Note that $\mathcal{C}(X)/{\sim}$ consists of 2 classes whose supports are $Y=\{1,2,4,5\}$ and $Z=\{1,3,5,6\}$ . For any $\theta\in\mathcal{AM}(X)$ , there are 2 possibilities for the corresponding isomorphisms $\lambda_1$ and $\lambda_2$ between the supports: either $\lambda_1\,{:}\,Y\to Y$ and $\lambda_2\,{:}\,Z\to Z$ , or $\lambda_1\,{:}\,Y\to Z$ and $\lambda_2\,{:}\,Z\to Y$ . In the former case $\lambda_1={\rm{id}}_Y$ and $\lambda_2={\rm{id}}_Z$ , and in the letter case $\lambda_2=\lambda^{-1}_1$ , where $\lambda_1$ maps an element $y\in Y$ to the element $z\in Z$ which is symmetric to y with respect to the vertical line passing through the vertices 1 and 5. In both cases, $\lambda_1$ and $\lambda_2$ are the restrictions of an automorphism of X to Y and Z, respectively.

4. Sets of length one

4.1. Admissible bijections of B and crowns in X

Before proceeding to the case $l(X)=1$ , we will prove a useful fact which holds for X of an arbitrary length.

Definition 4.1. Let n be an integer greater than 1. By a weak n-crownwe mean a poset $P=\{x_1,\dots,x_{n},y_1,\dots,y_{n}\}$ where

(11) \begin{align} x_i<y_i\ {for\ all}\ 1\le i\le n,\ x_{i+1}<y_i\ {for\ all}\ 1\le i\le n-1\ {and}\ x_1<y_{n}.\end{align}

An n-crownis a weak n-crown which has no other pairs of distinct comparable elements except (11). It is thus fully determined by n up to an isomorphism and will be denoted by ${\rm{Cr}}_n$ . A poset P is called a weak crown(resp. crown), if it is a weak n-crown (resp. n-crown) for some $n\ge 2$ . We say that a poset X has a weak crown (resp. crown) if there is a subset $Y\subseteq X$ which is a weak crown (resp. crown) under the induced partial order.

Posets without crowns are known to satisfy some “good” properties [Reference Dokuchaev and Novikov10, Reference Dräxler11, Reference Rival19].

Lemma 4.2. Let $\theta\in\mathcal{M}(X)$ . Then $\theta\in\mathcal{AM}(X)$ if and only if (2) holds for any $z\in X$ and any weak crown $\Gamma\,{:}\,u_0,\dots,u_m=u_0$ in X.

Proof. The “only if” part is obvious. For the “if” part take any closed semiwalk $\Gamma\,{:}\,u_0,u_1,\dots,u_m=u_0$ and $z\in X$ . Let $0\le k<k+l\le m$ such that $u_k<u_{k+1}<\dots<u_{k+l}$ . We define $\Gamma'\,{:}\,u_0,\dots,u_k,u_{k+l},\dots,u_m=u_0$ . By Corollary 3.2 equality (2) holds for $\Gamma$ if and only if it holds for $\Gamma'$ . The same is true for any $\Gamma'$ obtained from $\Gamma$ by removing intermediate terms in a decreasing sequence of consecutive vertices. Thus, doing so for all maximal sequences in $\Gamma$ we finally get $\Gamma'$ whose vertices form either a sequence $x<y>z$ or a weak crown. However, the case $\Gamma'\,{:}\,x<y>z$ can be ignored, because (2) always holds for such $\Gamma'$ as shown in the proof of [Reference Fornaroli, Khrypchenko and Santulo13, Lemma 5.13].