PatternAtlas

Sketches

Notes on the Powerset Lattice

A set of visual notes illustrating key ideas on powerset lattice

Cover page showing a powerset lattice of 3 elements Powerset Lattice A powerset lattice consists of all possible subsets of a set arranged in order of their containment within each other. Slide showing powerset lattice of 2 elements. The lattice can be thought of as a diamond with nodes at each of its 4 vertices. Copyright @patternatlas Powerset Lattice of Two Elements Let us start simple and start familiarizing ourselves with the subsets of a 2 element set. Shows the visualization of various subsets of the set {a, b}. These are: {}, {a}, {b}, {a,b}. Copyright @patternatlas Powerset Lattice of 2 Elements II We can arrange these subsets by the order of their containment as a lattice. Shows two visualizations of the powerset lattice. One denoting it using letters {a, b} and the other using balls of different colors. Copyright @patternatlas With the above concepts in mind, let us enumerate some powerset lattices. Here is the powerset lattice for one element. Shows the lattice {} ≤ {a}. Copyright @patternatlas As we have seen, the powerset lattice on two elements looks like this. Shows the lattice {} ≤ {a} || {b} ≤ {a, b}. Copyright @patternatlas Powerset lattice of 3 elements resembles a cube in its outline. Shows the lattice {} ≤ {a} || {b} || {c} ≤ {a, b} || {b, c} || {a, || {a, b, c}. Copyright @patternatlas And here is the hypercube shaped powerset lattice of four elements. The lattice 𝒫₄ shown can be described as: {} ≤ {a} || {b} || {c} || {d} ≤ {a, b} || {ac, } || {b, c} || {a, d} || {b, d} || {c, d} ≤ {a, b, c } || {a, b, d} || {a, c, d} || {b, c, d} ≤ {a, b, c, d} Copyright @patternatlas Powerset lattices with elements 1 to 4. Showcases all the lattices that have been introduced so far Copyright @patternatlas