Mathematics | Partial Orders and Lattices

Relations can be used to order some or all the elements of a set. For instance, the set of Natural numbers is ordered by the relation $\leq$ such that for every ordered pair $(x,y)$ in the relation, the natural number $x$ comes before the natural number $y$ unless both are equal.

Formally,

“A relation $R$ on set $A$ is called a partial ordering or partial order if it is reflexive, anti-symmetric and transitive. A set $A$ together with a partial ordering $R$ is called a partially ordered set or poset. The poset is denoted as $(S,R)$ .”

Example – Show that the inclusion relation $\subseteq$ is a partial ordering on the power set of a set $A$ .
Solution – Since every set $S\subseteq S$ , $\subseteq$ is reflexive. If $S\subseteq R$ and $R\subseteq S$ then $R = S$ , which means $\subseteq$ is anti-symmetric. It is transitive as $R\subseteq S$ and $S\subseteq T$ implies $R\subseteq T$ .
Hence, $\subseteq$ is a partial ordering on $P(S)$ , and $(P(S), \subseteq)$ is a poset.

Important Note : The symbol $\preceq$ is used to denote the relation in any poset. The notation $a \prec b$ is used to denote $a \preceq b$ but $a \neq b$ .

Comparability :

Let $a$ and $b$ be the elements of a poset $(S, \preceq)$ , then $a$ and $b$ are said to comparable if either $a \preceq b$ or $b \preceq a$ . Otherwise, $a$ and $b$ are said to be incomparable.

Example – In the poset $(Z^+, \mid)$ (where $Z^+$ is the set of all positive integers and $\mid$ is the divides relation) are the integers 3 and 9 comparable? Are 7 and 10 comparable?
Solution – 3 and 9 are comparable since $3 | 9$ i.e. 3 divides 9. But 7 and 10 are not comparable since $7 \nmid 10$ and $10 \nmid 7$ .

Total Order :

It is possible in a poset that for two elements $a$ and $b$ neither $a\preceq b$ nor $b\preceq a$ i.e. the elements $a$ and $b$ are incomparable. But in some cases, such as the poset $(Z^+, \leq)$ , every element is comparable to every other element.

A poset $(S, \preceq)$ is called totally ordered if every two elements of $S$ are comparable.
$\preceq$ is called a total order. A totally ordered set is also called a chain.

Hasse Diagrams :

A partial order, being a relation, can be represented by a di-graph. But most of the edges do not need to be shown since it would be redundant.
For instance, we know that every partial order is reflexive, so it is redundant to show the self-loops on every element of the set on which the partial order is defined.
Every partial order is transitive, so all edges denoting transitivity can be removed.
The directions on the edges can be ignored if all edges are presumed to have only one possible direction, conventionally upwards.
In general, a partial order on a finite set can be represented using the following procedure –

Remove all self-loops from all the vertices. This removes all edges showing reflexivity.
Remove all edges which are present due to transitivity i.e. if $(a,b)$ and $(b,c)$
are in the partial order, then remove the edge $(a,c)$ . Furthermore if $(c,d)$ is in the partial order, then remove the edge $(a,d)$ .
Arrange all edges such that the initial vertex is below the terminal vertex.
Remove all arrows on the directed edges, since all edges point upwards.

For example, the poset $(\{1, 2, 3, 4\}, \leq)$ would be converted to a Hasse diagram like –

The last figure in the above diagram contains sufficient information to find the partial ordering. This diagram is called a Hasse Diagram.

Extremums in Posets : Elements of posets that have certain extremal properties are important for many applications.

Maximal Elements- An element $a$ in the poset is said to be maximal if there is no element $b$ in the poset such that $a \prec b$ .
Minimal Elements- An element $a$ in the poset is said to be minimal if there is no element $b$ in the poset such that $b \prec a$ .

Maximal and Minimal elements are easy to find in Hasse diagrams. They are the topmost and bottommost elements respectively.
For example, in the hasse diagram described above, “1” is the minimal element and “4” is the maximal element. Since maximal and minimal are unique, they are also the greatest and least element of the poset.

Important Note : If the maximal or minimal element is unique, it is called the greatest or least element of the poset respectively.

Bounds in Posets :

It is somtimes possible to find an element that is greater than or equal to all the elements in a subset $A$ of poset $(S, \preceq)$ . Such an element is called the upper bound of $A$ . Similarly, we can also find the lower bound of $A$ .
These bounds can be further constrained to get the least upper bound and the greatest lower bound. These bounds are elements which are less than or greater than all the other upper bounds or lower bounds respectively.

Example – Find the least upper bound and greatest lower bound of the following subsets- $\{b, c\}$ , $\{g, e, a\}$ , $\{e, f\}$ .

Solution – For the set $\{b, c\}$
The upper bounds are – $e, f, h, i$ . So the least upper bound is $e$ .
The lower bounds are – $a$ . So the greatest lower bound is $a$ .
For the set $\{g, e, a\}$
The upper bounds are – $h$ . So the least upper bound is $h$ .
The lower bounds are – $a$ . So the greatest lower bound is $a$ .
For the set $\{e, f\}$
The upper bounds are – $f, h, i$ . So the least upper bound is $f$ .
The lower bounds are – $e, c, b, a$ . So the greatest lower bound is $e$ .

Lattices – A Poset in which every pair of elements has both, a least upper bound and a greatest
lower bound is called a lattice.
There are two binary operations defined for lattices –

Join – The join of two elements is their least upper bound. It is denoted by $\vee$ , not to be confused with disjunction.
Meet – The meet of two elements is their greatest lower bound. It is denoted by $\wedge$ , not to be confused with conjunction.

Sub Lattice – A sublattice of lattice $L$ is a subset $S\subseteq L$ such that if $a,b \in S$ , $a\wedge b \in S$ and $a\vee b \in S$ .

Identities for join and meet –

$\begin{flalign*} &\bullet\: x\wedge x = x\:&&and\: x\vee x = x \\ &\bullet\: x\wedge y = y\wedge x\:&&and\: x\vee y = y\vee x \\ &\bullet\: (x\wedge y)\wedge z = x\wedge (y\wedge z)\:&&and\: (x\vee y)\vee z = x\vee (y\vee z) \\ &\bullet\: x\wedge (y\vee x) = x\:&&and\: x\vee (y\wedge x) = x \\ \end{flalign*}$

Distributive laws may or may not hold true for a lattice :

$x \wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$
$x \vee (y\wedge z) = (x\vee y) \wedge (x\vee z)$

Note – A lattice is called a distributive lattice if the distributive laws hold for it.

But Semidistributive laws hold true for all lattices :

$x \wedge (y\vee z) \geq (x\wedge y) \vee (x\wedge z)$
$x \vee (y\wedge z) \leq (x\vee y) \wedge (x\vee z)$

Two important properties of Distributive Lattices –

In any distributive lattice $a\wedge y = a \wedge x$ and $a\vee y = a \vee x$ together imply that $x = y$ .
If $a\wedge x = O$ and $a\vee x = I$ , where $O$ and $I$ are the least and greatest element of lattice, then $a$ and $x$ are said to be a complementary pair. $O$ and $I$ are a trivially complementary pair.