I am asked to prove that every chain is a distributive lattice.
Is it true that every chain is a lattice?
I am told that a chain is a poset where we can compare any two elements.
A lattice is a poset where every subset has a lub and a gld. I don't understand how then every chain is a lattice.
what if we take two elements in a chain that do not have a lub, then how can we even talk about distributive of meets over joins?
Best Answer
To prove that every chain $\langle P, \leqslant\rangle$ is a lattice, fix some $a, b \in P$ and w.l.o.g assume that $a \leqslant b$. From reflexivity of $\leqslant$ it follows that $a \leqslant a$, hence $a$ is a lower bound of the set $\{a,b\}$. To prove that it is the greatest lower bound note that if some $c \in P$ is another lower bound of $\{a,b\}$ then by the definition of a lower bound we have $c \leqslant a$. It means that $a$ is the greatest lower bound of $\{a,b\}$. Same reasoning shows that $b$ is the least upper bound of $\{a,b\}$.
To prove that every chain $\langle P, \leqslant\rangle$ is distributive, you should just consider all possible relations between three arbitrary elements $a, b, c \in P$ and check that distributive identity holds.
For example, let $a \leqslant b \leqslant c$, hence $a \wedge b = a \wedge c = a$ and $b \vee c = c$, so $$a \wedge (b \vee c) = a \wedge c = a = a \vee a = (a \wedge b) \vee (a \wedge c).$$