Actions on posets

Preorders and monoid actions are deeply related. Starting with a monoid action, we can get a preorder. This preorder then determines the lattice of subobjects of the monoid action. In the other direction, when considering actions on preorders we can get additional types of monoid actions. This leads to the following ontology of posetal actions, all of which form monoids:

Typically, when considering the endomorphisms monoid $End(X)$ of an object $X$ of a category $C$ we can create a full theory of actions, but posets are different because their strong link with actions. In order to continue further, it is useful to add a posetal structure on the hom classes of posetal transformations.

Definition. let $(P, \subseteq), (Q, \subseteq)$ be partially ordered sets then we can partially order the hom class $Hom(P,Q)$ so that $f, g \in Hom(P,Q)$ have $f \subseteq g$ iff $\forall x : f(x) \subseteq g(x)$.

Let $(P, \subseteq)$ be a poset, then this can even be applied to $T_P$ the set of all transformations on $P$. Then we have taht a transformation $f$ is increasing iff $id_P \subseteq f$ and it is decreasing iff $f \subseteq id_P$. It follows that increasing transformations are the principal filter of the identity function $id_P$ in the pointwise ordering, and decreasing transformations are its principal filter. The only transformation that is both increasing and decreasing is therefore the identity $id_P$.

Theorem. The monoid of increasing actions on a poset is R-trivial.

Proof. Suppose that we have $f,g$ that are R-related then we have that $af = g$ and that $bg = f$. By the fact that each action is increasing we have that $\forall x : x \subseteq f(x) \subseteq a(f(x)) = g(x)$. In the other direction, we have that $\forall x : x \subseteq g(x) \subseteq b(g(x)) = f(x)$. The statement that $f(x) \subseteq g(x)$ and $g(x) \subseteq f(x)$ contradicts antisymmetry, so $f,g$ cannot be R-related. $\square$

Although it is the case that increasing actions on a poset are R-trivial, we cannot get that they are L-trivial in general, however, in the restricted case of increasing monotone actions then we can get L-triviality.

Theorem. The monoid of increasing monotone actions is D-trivial

Proof. Suppose that $f \subseteq_L g$ then $\exists a : fa = g$. We have that $\forall x : x \subseteq a(x) \subseteq f(a(x))$. By monotonicity, $x \subseteq a(x)$ implies $f(x) \subseteq f(a(x))$ now since $f(a(x)) = g(x)$ this implies $f(x) \subseteq g(x)$. So therefore, if $f,g$ are in the same L class then we have $f(x) \subseteq g(x)$ and $g(x) \subseteq f(x)$ which contradicts antisymmetry. Therefore, the monoid of increasing monotone actions is L-trivial. It is already R-trivial, so if it is both L-trivial and R-trivial then it is D-trivial. $\square$

One corollary that is worth mentioning, is that the poset of monoid actions is group-free. This distinguishes increasing actions from monotone ones which can include the group of automorphisms.

Corollary. The monoid of increasing actions is group-free

Proof. R-trivial $\implies$ H-trivial $\implies$ group-free $\square$

Let $C$ be the category of partial orders and monotone maps. Then we can easily make the following observation about the group of automorphisms $Aut(P)$ of a finite poset $P$.

Theorem. Let $P$ be a finite poset. Then $Aut(P)$ is never increasing/decreasing.

Proof. We can rank the elements of $P$ in the following form: $R(x)$ is equal to the maximum height of the principal ideal of $x$. Then if we have $x \subset y$ we must have that $R(x) \lt R(y)$. Permutable elements must have the same invariant rank, therefore they must be incomparable. $\square$

It is wrong to assume that $Aut(P)$ is never increasing/decreasing in general because it may not always be possible to rank elements of a poset. In particular, there are order-automorphisms of the dense total order $(\mathbb{Q}, \subseteq)$ because no well-ordered ranking can be established on it.

Corollary. The orbits of the group action of a finite partially ordered set $Aut(P)$ are an equivalence subrelation of the partition of $P$ into co-connected components.

The automorphism group of a finite weak order is an orbit symmetric group, and the orbits of the group action coincide with the connected components of the complement of the comparability graph.