Is a single element a chain or an anti-chain?

Question

Is a single element in a Hasse Diagram a chion or an anti-chain or both?

• As a Hasse Diagram contains reflexive elements, it is compararble to itself. So it is a chain.
• But I read somwhere that a single element is also an anti-chain

How is it an anti-chain if the Hasse diagram does not contain any other element to which it might be incomparable?

Answer 1

A subset $A$ of a partial order $\langle P,\le\rangle$ is an antichain if and only if there do not exist $a,b\in A$ such that $a\le b$ and $a\ne b$. In other words, what keeps a set $A\subseteq P$ from being an antichain is having two different elements that are related by the order $\le$. If $A=\{x\}$, then $A$ doesn’t have two different elements at all, so it certainly doesn’t have two different elements that are related by the order, and it must therefore be an antichain.

Perhaps you’ve seen the following equivalent definition of an antichain: $A\subseteq P$ is an antichain if and only if for all $a,b\in A$, $a\le b$ implies that $a=b$. If you use this definition, it is if anything even clearer that $\{x\}$ is an antichain for each $x\in P$: if $a,b\in\{x\}$, then $a=x$ and $b=x$, so $a=b$ whether or not $a\le b$ (though of course it’s true in this case that $a\le b$).

Answer 2

An anti-chain is a subset $A$ of a poset $P$ such that, if $p, q \in A$ and $p \neq q$, then $p \not\le q$ and $q \not \le p$. If $P$ is a singleton, then this is vacuously true.

Vacuous truth can occur when we have a "for all" statement or an "if-then" statement. An "if-then" statement is vacuously true if the premise of the if-then statement is never true. For example,

If $x \in \Bbb{R}$ such that $x^2 + x + 1 = 0$, then $x = 6$ and $x = -3$.

is a vacuously true statement. Even though the conclusion can never be true, the premise is always false. It is a logical convention to call this implication true (but we consider its truth "vacuous", because it's only true due to an absence of a counterexample).

You can sort of think of it as being harmless to consider it true. A true if-then statement, whenever the premise is true, allows us to safely conclude the conclusion. If the premise is never true, we never are allowed to conclude the conclusion (at least, using this statement).

Similarly, a "for-all" statement is vacuously true if we are considering all things in a set that turns out to be empty. For example,

For all unbounded continuous functions on $[0, 1]$, there exists some $c \in [0, 1]$ such that $f(c) = 12$.

is another vacuously true statement, simply because every continuous function on $[0, 1]$ is bounded. Again, it's harmless to consider this true, simply because we will never get an unbounded continuous function on $[0, 1]$, so the conclusion is ultimately irrelevant.

So, back to anti-chains, if we have $A \subseteq P$ is a singleton (or even empty!), we can never find $p, q \in A$ such that $p \neq q$. This is vacuous truth, and hence $A$ is indeed an anti-chain.