Question

To determine an example of an infinite lattice with both a least and a greatest element.

Short answer

The lattice \((P(S), \subseteq )\) with both a least and a greatest element.

Step-by-step solution

Given data

Lattice is an infinite.

Concept used of partially ordered set

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and\((y,x)\)in R implies\(x = y\)(anti-symmetry),\((x,y)\)and\((y,z)\)in R implies\((x,z)\)is in\({\rm{R}}\)(transitivity).

Find lattice with both a least and a greatest element

We have to find an example of an infinite lattice which has both a least and greatest element.If \(S\) is an infinite set, then \(P(S)\) the set all subsets of \(S\), is finite and it is a lattice with respect to the set inclusion ( \( \subseteq \) ).Therefore, \((P(S), \subseteq )\) is a lattice.

The empty set \{\} is the least element since it is subset of every set and \(S\) is the greatest element of the lattice \((P(S),S)\) since it contained every element of \(P(S)\).

Hence, the lattice \((P(S), \subseteq )\) with both a least and a greatest element.