Every bounded lattice may or may not be finite.

Question

I want an example of a bounded lattice which is infinite with hasse diagram. As finite lattice is a lattice which surely contains greatest and least element then what is infinite lattice. Please elaborate that also.

Answer 1

Let $0$, $x_i$ for $i$ natural, and $1$ be the elements of the lattice. $0<x_i<1$ for all $i$, and the $x_i$ are pairwise incomparable. This is a bounded infinite lattice that has a Hasse diagram.

Answer 2

For any infinite set $X$,

1. $\mathcal P(X)$, the lattice of subsets of $X$, is bounded with bounds $\varnothing$ and $X$,

2. $\Pi(X)$, the lattice of equivalence relations on $X$, is bounded with bounds $\Delta_X = \{(x,x):x\in X\}$ and $\nabla_X = \{(x,y):x,y\in X\}$,