2026-08-01 Bounds for Ordered Sets

The essential idea from the previous post on orders is that comparing the elements of a set gives rise to relations between them and adding a “sense” (a measure that quantifies the relation mathematically) to the relation allows for the ordering of elements within the set. Taking only those ordered sets where elements can be related to themselves (irreflexive) and where not every element is related to another in the set (partial), taking subsets of such posets and checking if any notable properties emerge from them is interesting exercise, so let’s start there.

Subsets of ordered sets #

Take a subset $Y$ of an partially ordered set $X$ with relation $R$ on it. In $X$ :

• a lower bound is an element $x$ where $x R y, \forall y \in Y$ (x bounds Y from below)
• an upper bound is an element $x$ where $y R x, \forall y \in Y$ (x bounds Y from above)

The set of all lower bounds is defined as:

and the set of all upper bounds as:

(the set of all values in X that bound the subset Y from above)

Note that all of these arises from nothing more than taking a subset of the original partially set and using the relation $R$ to relate members of the original set $X$ and the subsets $Y$ . Because $X$ is an ordered set using a binary relation $R$ , the sets of bounds are also ordered in two directions - lower and upper.

When the set of upper bounds Y$^u$ has a least element, it is the least upper bound of $Y$ which is given by $x$ :

$$(\forall x' \in X [((\forall y \in Y) y R x' \iff x R x')])$$

Considering the dual, the set of lower bounds Y$^l$ has a greatest element or the greatest lower bound and is given by $x$ :

$$(\forall x' \in X [((\forall y \in Y) x' R y \iff x' R x)])$$

In literature, the least upper bound is the supremum (written as sup $Y$ ) of the subset $Y$ and the greatest lower bound is the infimum of Y
(written as inf $Y$ )

Important points to note are:

• A set Y need not have a supremum or an infimum.

  • E.g. Let X = {a, b, c, d} be a poset with:

    $R = \{a, c\}, \{a, d\}, \{b, c\}, \{b, d\}$ (neither c R d nor d R c holds).

  Take Y = {a, b}. To calculate the set of upper bounds for Y, $x \in X$ must be $y R x$ i.e. for each $y \in Y$ :

  y	relation	x
  a	a R c	c
  a	a R d	d
  b	b R c	c
  b	b R d	d

  S^u = {c, d}

  To get only one element out of $S^u$ requires a comparision between c and d but since R does not hold for both, there’s no least element or supremum in {c, d}.

• A supremum or an infimum, if either exist, are always unique. Looking at the definition above, if x and x’ are both upper bounds in $X$ , then we must have $x R x'$ and $x' R x$ which is only possible when x’ = x in a poset (the antisymmetry property).

• The supremum or the infimum of a set may or may not belong to the set itself.

  • The closed interval [0, 1] of reals has a sup of 1 which is contained in the set itself.
  • The open interval (0, 1) of reals again has a sup of 1 but this is not contained within the set {0,1}.

An poset X has a bottom element if there exists $\bot \in X$ (called bottom) with the property that $\bot R x$ for all x $\in X$ . The dual element in X is a top element which if exists is defined as $x R \top$ for all $x \in x$ . For the set of upper bounds Y$^u$ , the least upper bound or supremum $= \top$ and the set of lower bounds Y$_u$ , the greatest lower bound or infimum $= \bot$ .

Lattices #

So far, the definitions of subsets of posets have meant any subset of a poset $X$ . Now if the definition were to narrowed down to every two-element (or doubleton) subset of $X$ , then a structure called a lattice emerges from $X$ only if:

• $\forall x, y \in X$ , the least upper bound exists. This is called a join and denoted by $x \vee y$
• $\forall x, y \in X$ , the greatest upper bound exists. This is called a meet and denoted by $x \wedge y$

A poset $X$ is a join-semilattice (or upper-semilattice) in which $\forall x, y \in X$ , $x \vee y$ exists. The dual holds for meet-s semilattices (lower-semilattices) i.e. $\forall x, y \in X$ , $x \wedge y$ .

For a subsets $S \subseteq X$ :

• $\bigvee S$ is the least upper bound or join of subset $S$
• $\bigwedge S$ is the greatest lower bound or meet of subset $S$

If both $\bigvee S$ , $\bigwedge S$ exist for all $S \subseteq X$ , the $X$ is a complete lattice.

Filters #

A principal filter or principal up-set on a poset $X$ generated by an element $y$ is defined as:

$$\uparrow y = \{ z \in X | y R z \}$$

$\uparrow y$ is everything in the poset that is “at” or “above” y (at because y R y by definition).

The set of upper bounds of two element set, $\{x, y\}^{u} is given by$ y$ since x R y, nothing “below” $Y$ can be an upper bound. Since the least element of $\uparrow y$ is y, $x \vee y = y$ .

A principal down-set on a poset $X$ generated by an element $z$ is defined as:

$$\downarrow x = \{ z \in X | x R z \}$$

$\downarrow x$ is everything in the poset that is “at” or “below” x (at because again x R x by definition).

The set of lower bounds of two element set, $\{x, y\}^{l}$ is given by $\downarrow x$ as nothing “below” x can be a lower bound. Since the least element of $x is x, $x \wedge x = x$ .

Hasse diagrams #

This far, diagrams for definitions have shown abstract versions of posets, their subsets, upper/lower bounds. One representation of specific posets are Hasse diagrams. A Hasse diagram for a poset $X$ with $x, y \in X$ is done as follows:

1. Each $x \in X$ is represented by a small circle.
2. For each pair x $\lessdot$ y (y covers x i.e. y immediately succeeds x when ordered using $R$ ), a line from x to y is drawn
3. 1. and 2. must adhere to the following:
   • when x $R$ y, the circle depicting x is lower than the one
     depicting y
   • the lines joining circles may cross each other but the circles depicting elements much never intersect lines.

This Hasse diagram is a representation of poset $X = \{(a, d), (b, e), (b, d), (b, f), (c, f), (c, g), (d, h), (e, h), (e, i), (f, i), (g, i)\}$

Moving upward from an circle depicting an element shows the transitive relations e.g. b R e R h $\implies$ b R h. Elements e and d are not ordered by $R$ ($d \hspace{0.2em}|| \hspace{0.2em} e$ ). The reflexivity of elements is implied.

Lattices as an Algebraic Structure #

A mathematical or a computational entity has an algebraic structure when it is comprised of:

• a set of elements
• a finite collection of operations on its elements
• and a finite set of identities (axioms) which hold for all possible elements of the structure.

Sets themselves have identities that any set must follow. As ordered sets with other features, lattices are algebraic structures too,a denoted by $⟨L; \vee, \wedge⟩$

The following rules are equivalent for all lattices from poset $X$ , where $x, y\in X$ :

1. x R y
2. x $\vee$ y = y
3. x $\wedge$ y = x

and give rise to the following axioms with $x, y, z \in X$ :

• (x $\vee$ y) $\vee$ z = x $\vee$ (y $\vee$ z) (associative)
• x $\vee$ y = y $\wedge$ x (commutative)
• x $\vee$ x = x (idempotency)
• x $\vee$ (x $\wedge$ y)= x (absorption)

The four axioms hold for their dual versions where $\vee$ and $\wedge$ are exchanged.

Next #

Chapter 2 of the Lattices and Order has far more details on the topic of lattices if you are interested. A thorough understanding of these definitions should, hopefully, be enough to dig through how a lattice structure has been added to OCaml’s type system to build OxCaml in the next post.

It’s thanks to Claude credits that I could come up with the illustrative diagrams very easily.

2026-07-31 Putting Everything in Order

A paper I have been reading through on recently (OxCaml) does some very clever things with the OCaml type system. The basic idea is to wrap OCaml function types with qualifiers to ensure memory access safety statically from within the type system.

I wanted to write down notes on parts of the paper but then thought of a different idea - start with the underlying structure that has been used to add qualifiers to the type system and go from there. And for that, I thought I would start, as much as possible, from the beginning with relations and orders.

The Basics - Relations #

To understand what a relation is, let’s start with a set. Relying on the definition of a set to be a collection of objects, a relation can be thought of as a:

• function between sets, for example:

  • A as a set of numbers from 1 to 26 and
  • B as the set of all letters in the English alphabet,
  • the relation “is the nth letter in the alphabet” written as a function: $f(x) = y$ where $x \in A$ and $y \in B$

• sets of elements, where:

  • taking A as a set of numbers from 1 to 26 and
  • B as the set of all letters in the English alphabet,
  • the relation “is the nth letter in the alphabet” is denoted by R and consists of the set of pairs defined as: $(x, y) \in R$ where $x \in A$ and $y \in B$ .

Using the second definition, for any two sets X and Y, denoting X x Y as the set of all possible pairs with $x \in X$ , $y \in Y$ , a binary relation R is its subset i.e. the set containing pairs for R holds for $x,y$ . When both $x, y \in X$ , R is a relation on X itself.

Properties of Relations #

Using the notation $xRy$ for $(x,y) \in R$ , relations on a set X for all $x,y,z \in X$ can be characterized as:

Type	Notation	Note
empty	$\neg(xRy)$	$\neg$ denotes negation; in an empty set, no relation can hold
reflexive	$xRx$
irreflexive	$\neg(xRx$ )
identity	$xRy \to x = y$
transitive	$xRy \land yRz \to xRz$	$\land$ denotes $and$
symmetric	$xRy \to yRx$
antisymmetric	$xRy \land yRx \to x=y$
asymmetric	$xRy \to \neg(yRx)$

It always helps to work through examples for dry mathematical definitions, so let’s see some for each property listed above. Given $x,y,z \in$ a set $X$ :

• empty ($\neg(xRy))$ :

  • R = $\varnothing$ and $X = {1,2,3,4,5}$ denotes R has no pairs. All of the other properties become vacuously true (hold for the empty set) except for the reflexive one. Since R is empty, there isn’t a way to relate every $x$ to itself.

  • The one subtlety is that if both R = $\varnothing$ and $X = \varnothing$ , then the reflexive property holds along with all others (all elements of an empty set are related to themselves via an empty relation).

• reflexive ($xRx$ ):

  • The relation $\leq$ on the set of integers where R = ${(x, x) | \hspace{0.5em} x \leq x}$ (E.g. 1 $\leq$ 1, 2 $\leq$ 2, …)

  • The relation “is reachable from” for nodes in a graph. R = ${(x, x) | \hspace{0.5em} \text{any node is reachable from itself}}$

• irreflexive ($\neg(xRx$ )): The relation $\le$ on the set of integers $X$ .

  • R = ${(x, x) | \hspace{0.5em} \neg(x \lt x)}$ = $\varnothing$ = $neg(1 \lt 1, 2 \lt 2, ...)$

  • R = ${(x, y) | \hspace{0.5em} x \subsetneq y}$ where $x,y$ are sets and elements of the powerset of set X. ($\subsetneq$ stands for “is a subset of and not equal to” and denotes a strict subset).

• identity ($xRy \to x = y$ ):

  • The relation “is equal to” on the set of natural numbers $X$ R = ${(x, y) | \hspace{0.5em} x = y }$

  • The relation “the result from an identity function” where R = $y = f(x) = {(x, y) | \hspace{0.5em} x = y}$

• transitive ($xRy \land yRz \to xRz$ ):

  • The relation $\leq$ on the set of integers where R = ${(x, x) | x \leq x}$ (1 $\leq$ 2 and 2 $\leq$ 3 $\to$ 1 $\leq$ 3)

  • The relation “is an ancestor of” in a graph whose nodes form a set $xRy$ and $yRz$ $\to$ $xRz$ i.e. node x is an ancestor to both y and z

• symmetric ($xRy \to yRx$ ):

  • For X = $\mathbb{R} \backslash {0}$ i.e. the set of reals excluding 0, multiplicative inverses i.e. $R = {(x, y) | \hspace{0.5em} x = 1/y}$

  • For X = $\mathbb{R}$ only, additive inverses i.e. $R = ${(x, y) | x = -y}$

  • For X be the set of all nodes in a graph with the depth of a node defined from its root. Then $R = {(x, y) | \hspace{0.5em} depth(a) = depth(b)}$

  are all symmetric

• antisymmetric ($xRy \land yRx \to x=y$ ):

  • For $X$ as the set of positive integers, xRy where R is “divides”. If x were to divide y and y x, then the only possibility is that x = y.

  • R = ${(x, y) | \hspace{0.5em} x \subset y}$ where $x,y$ are sets and elements of the powerset of set X. ($\subset$ stands for “is a subset of”). Only if $x = y$ can $xRy \land yRx$ hold

• asymmetric ($xRy \to \neg(yRx)$ ):

  • For X be the set of all nodes in a graph with R defined as “is the parent of”. R = ${(x, y) | \hspace{0.5em} \text{node x is node y's parent}}$ Alternatively: R = ${(x, x) | \hspace{0.5em} \text{node x cannot be its own parent}}$

  • The relation $\lt$ on the set of integers where R = ${(x, y) | x \lt y}$

There are two important points for each property described above where each:

• holds for a relation R if and only if it holds of its converse R$^{op}$ , defined by $xRy \iff yR^{op}x$ (courtesy of the duality principle)

• extends to Boolean combinations of the above properties i.e. those formed using the boolean ‘and’ ($\land$ ), ‘or’ ($\lor$ ) and ‘not’ ($\neg$ ).

The Sense in Relations #

With sets as a collections of elements and relations between such elements defined above, an idea of how we can use such relations to position elements within the set comes from the definitions of the relations themselves.

The original idea of using relations to determine relative positions of elements i.e. an order between them comes from a paper by Bertrand Russell titled “On the Notion of Order”. The crux of the paper is as follows:

A casual collection of terms may be ordered by counting, in which case they are correlated with the integers; by speech, in which case they are correlated with a series of times; or by writing, in which case they correlated with a series of places. But the order arises, in each case, from the intrinsic order of the integers, the times, or the places respectively. These have an order independent of our caprice-they form what I shall call independent or self-sufficient series. The casual terms correlated with them form, on the contrary, only a series by correlation. Series by correlation are generated from self-sufficient series as follows: If there be a self-sufficient series A, B, C, D, . . . a collection of terms $\alpha$ , $\beta$ … and a specific relation R which subsists between $alpha$ and A, and $\beta$ and B, etc., but not between $\alpha$ and B or C or D or etc. (with similar exclusions for $\beta$ …), then $\alpha$ , $\beta$ … acquire, by correlation with A, B, C, D, the order which belongs intrinsically to A, B, C, D. Thus all orders by correlation are logically dependent upon intrinsic orders. The latter alone will be considered in what follows.

Order depends fundamentally upon relations having what mathematicians call sense, i.e., such that the relation of A to B is different from that of B to A. Such are east and west, greater and less, before and after, etc. But if order is to arise, another condition is necessary. It must be possible for the same relation with opposite senses to attach to a given term. This excludes such relations as occupation of a place or a time. For though a time may be occupied by an event, there is nothing which the time itself can occupy; and similarly as regards a place. Where both conditions are satisfied, we in general have an order. That is, if there be any relation R, having two senses R$_{1}$ , R$_{2}$ ; and if a term B have the relation R$_{1}$ to A, while it has the relation R$_{2}$ to C, then B is between A and C, and the three terms have the, order ABC or CBA. Thus these two conditions are necessary for an intrinsic order of three terms, and become sufficient if we add that BR$_{1}$ A, BR$_{2}$ C are to imply the denial (emphasis mine) of AR$_{1}$ C”

The idea is that a “collection of terms” (a set) with binary relations between its elements has an “intrinsic” order where if R is a relation on elements $x, y, z \in$ set $X$ with the property of being:

• asymmetric (($xRy \to \neg(yRx)$ ))
• transitive ($xRy \land yRz \to xRz$ )

then elements $x, y, z$ can be compared using relations such as “is derived from”, “is less than”, “is contained in”, “happened before (/after)”, “comes before”. The asymmetry ensures the transitivity of the relation occurs only in one direction, without which there would be no way to put elements from $X$ in any distinct, comparative order. This is the “sense” talked about in the paper and translates to the common sense notion of order we use in our thinking and everyday speech.

Types of Orders #

From the fundamental understanding of an order in the last section, let’s build the definitions by adding or slightly modifying properties of relations.

If we were to add:

• irreflexive ($\neg(xRx$ ))
• transitive ($xRy \land yRz \to xRz$ ) (remains the same)
• asymmetric (($xRy \to \neg(yRx)$ )) (remains the same)

then a relation $R$ on ($x, y, z \in$ ) set $X$ is a (strict) partial order. The “partial” signifies that the relation holds only for those elements in the set which can be related using $R$ implying that it is not necessary for each element in $X$ to be related to every other element. The “strict” ensures that elements in $X$ can only be related to other element, never to itself.

Modifying the above definition to:

• irreflexive ($xRx$ )
• transitive ($xRy \land yRz \to xRz$ ) (remains the same)
• asymmetric (($xRy \to \neg(yRx)$ )) antisymmetric ($xRy \land yRx \to x=y$ )

defines a weak partial order and is denoted by $(X, ≤)$ . The set with such a order is a (weakly) partially ordered set (or a poset). The “weak” signifies that elements can be related to themselves.

An interesting mathematical property is that a relation is asymmetric if and only if it is both antisymmetric and irreflexive. So the asymmetry property of the strict partial order subsumes both an irreflexive property (superfluous in the definition actually) and an antisymmetry property. Weakening the definition to make the relation reflexive means the asymmetry property can no longer hold i.e. now when $xRy$ , it is possible $yRx$ . With the addition of antisymmetry, this can only be true when $x = y$ . This ensures that two elements can only be related in “both directions” if they are the same, implying that the equivalent guarantees of the asymmetry property are maintained in the presence of reflexivity for weak partial orders.

To work through a very simple example of a poset, let’s take a set $S$ = {1,2,3} and its powerset P($S$ ) = {$\varnothing$ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1, 2, 3}}. With $R$ defined as $\subseteq$ i.e. subset, $R$ is:

• reflexive - a set is always its own subset ({1} $. {2} $\subseteq$ {2} in P($S$ ))
• transitive - any element S$_1$ within S is a subset of another element S$_2$ which in turn is the subset of another element S$_3$ ({1} $\subseteq$ {1,3} $\subseteq$ {1,2,3} $\implies$ {1} $\subseteq$ {1,2,3}; all elements is P($S$ ) are subsets of the maximal element {1,2,3})
• antisymmetric - every element S$_1$ of P($S$ ) is subset of another element S$_2$ ; if S$_2$ is a subset of S$_1$ , then S$_1$ = S$_2$ ({1,2} $\subseteq$ {1,2} and {1,2} $\subseteq$ {1,2} only when {1,2} = {1,2})

An poset $X$ is termed a chain or a linearly ordered set or a totally ordered set when any two elements of $X$ are comparable ($\forall x, y \in X, xRy \lor yRx$ ).

The book “Introduction to Lattices and Order” covers many more definitions and examples on orders. The next topic that arises from the study of relations and orders are lattices which I’ll write it up in my next blog post.

References:

• Notes on Lattices from a course on Algebraic Logic
• Copy of Lattices and Order