The following comes from Definition 2 in Pavel Pudlak, "A new proof of the congruence lattice representation theorem," *Algebra Universalis* **6** (1976), 269-275.


Let $X$ be a set. Let $F$ be a family of functions from $X$ to itself containing the identity map and closed under composition.


Define a binary relation on the family of one- or two-element subsets of $X$ as follows. Let $a,b,c,d\in X$. We will say that $\{a,b\}$ *dominates* $\{c,d\}$ if there are $n\in\mathbb N\_0$, $u\_0,\dots,u\_n\in X$, and $f\_1,\dots,f\_n\in F$ such that $u\_0=c$, $u\_n=d$, and $\{f\_i(a),f\_i(b)\}=\{u\_{i-1},u\_i\}$ for $i=1,\dots,n$.


Why is domination transitive?



Why Is Pudlak's relation on the family of one- or two-element subsets of a set transitive?
I don’t see what the problem is supposed to be; you just compose the functions witnessing the two domination relations in the obvious way:


Fix $c=u\_0,\dots,u\_n=d$ and $f\_1,\dots,f\_n\in F$ such that $\{f\_i(a),f\_i(b)\}=\{u\_{i-1},u\_i\}$ for each $i=1,\dots,n$.


Fix $e=v\_0,\dots,v\_m=f$ and $g\_1,\dots,g\_m\in F$ such that $\{g\_j(c),g\_j(d)\}=\{v\_{j-1},v\_j\}$ for each $j=1,\dots,m$.


Then the fact that $\{a,b\}$ dominates $\{e,f\}$ is witnessed by the sequence
$$e=v\_0=w\_{1,0},w\_{1,1},\dots,w\_{1,n}=v\_2=w\_{2,0},w\_{2,1},\dots,w\_{m,n}=v\_m=f,$$
where
$$w\_{j,i}=\begin{cases}g\_j(u\_i)&\text{if }g\_j(c)=v\_{j-1}\text{ and }g\_j(d)=v\_j,\\
g\_j(u\_{n-i})&\text{otherwise.}\end{cases}$$
We have $\{w\_{j,i-1},w\_{j,i}\}=\{h\_{j,i}(a),h\_{j,i}(b)\}$, where
$$h\_{j,i}=\begin{cases}g\_j\circ f\_i&\text{if }g\_j(c)=v\_{j-1}\text{ and }g\_j(d)=v\_j,\\
g\_j\circ f\_{n+1-i}&\text{otherwise}\end{cases}$$
is in $F$ as it is closed under composition.


