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

1. Let $C$ be any maximal chain and $A$ be any antichain. Then $A$ is covered by both the sets $Past(x)$ and $Future(x)$ for $x$ running over $C$ and where $Past(x):=\lbrace y\mid y \leq x\rbrace$ and $Future(x):=\lbrace y \mid x \leq y\rbrace$.

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

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