Question

Given is a locally finite countable connected poset which satisfies further the following properties:

1. Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is covered by both the sets ${\rm Past}(x)$ and ${\rm Future}(x)$ for $x$ running over $C$ i.e. $A \subset \bigcup_{x \in C} ({\rm Past}(x) \cup {\rm Future}(x))$ and where ${\rm Past}(x):=\lbrace y\mid y \leq x\rbrace$ and ${\rm Future}(x):=\lbrace y \mid x \leq y\rbrace$.

2. For any $x$, the intersection of ${\rm Past}(x)$ with any antichain is finite. Similarily for ${\rm Future}(x)$.

   Questions:

      1. Is the orbit of any point $x$ by an automorphism $f$ of the poset finite?
   
   
      2. Is the group of automorphisms of this poset countable?
   
   
      3. As a polish group, is the group of automorphisms of the poset locally compact?
   

   Thank you