# Order (relation)

In mathematics, an **order relation** is a relation on a set which generalises the notion of comparison between numbers and magnitudes, or inclusion between sets or algebraic structures.

Throughout the discussion of various forms of order, it is necessary to distinguish between a *strict* or *strong* form and a *weak* form of an order: the difference being that the weak form includes the possibility that the objects being compared are equal. We shall usually denote a general order by the traditional symbols < or > for the strict form and ≤ or ≥ for the weak form, but notations such as ,;
,;
, are also common. We also use the traditional notational convention that .

An *ordered set* is a pair (*X*,<) consisting of a set and an order relation.

## Partial order

The most general form of order is the (weak) **partial order**, a relation ≤ on a set satisfying:

*Reflexive*:*Antisymmetric*:*Transitive*:

The *strict* form < of an order satisfies the variant conditions:

*Irreflexive*:*Antisymmetric*:*Transitive*:

Weak and strict partial orders are equivalent via the following translations:

- if and only if or
- if and only if and

A reflexive and transitive relation is called a **preorder**. In a preorder the relation defined by is an equivalence relation, and the preorder gives rise to a partial order on the corresponding equivalence classes.

## Total order

A **total** or **linear order** is one which has the *trichotomy* property: for any *x*, *y* exactly one of the three statements , , holds.

## Associated concepts

If *a* ≤ *b* in an ordered set (*X*,<) then the *interval*

We say that *b* *covers* *a* if the interval : that is, there is no *x* strictly between *a* and *b*. We write or .

Let *S* be a subset of a ordered set (*X*,<). An *upper bound* for *S* is an element *U* of *X* such that for all elements . A *lower bound* for *S* is an element *L* of *X* such that for all elements . A set is *bounded* if it has both lower and upper bounds. In general a set need not have either an upper or a lower bound. A *directed set* is one in which any finite set has an upper bound.

The set of upper bounds for *S* is denoted *UB*(*S*); the set of lower bounds is *LB*(*S*).

A *supremum* for *S* is an upper bound which is less than or equal to any other upper bound for *S*; an *infimum* is a lower bound for *S* which is greater than or equal to any other lower bound for *S*. In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum. The supremum or infimum of *S*, if one exists, is unique

A *maximum* for *S* is an upper bound which is in *S*; a *minimum* for *S* is a lower bound which is in *S*. A maximum is necessarily a supremum, but a supremum for a set need not be a maximum (that is, need not be an element of the set); similarly an infimum need not be a minimum.

A maximum element for the whole set may be termed *top*, *one* or *true* and denoted by or **1**; a minimum element for the whole set may be termed *bottom*, *zero* or *false* and denoted or **0**. An ordered set with a **0** and **1** is *bounded*.

In a bounded order, an **atom** is an element that covers **0**.

An *antichain* is a subset of an ordered set in which no two elements are comparable. The *width* of a partially ordered set is the largest cardinality of an antichain.

A subset *S* of an ordered set *X* is *downward closed* or a *lower set* if it satisfies

Similarly, a subset *S* of an ordered set *X* is *upward closed* or an *upper set* if it satisfies

A (*Dedekind*) *cut* in an ordered set *X* is a pair (*A*,*B*) of subsets of *X* such that *B* is the set of upper bounds of *A* and *A* is the set of lower bounds of *B*: *B* = *UB*(*A*) and *A* = *LB*(*B*). We may equivalently define a cut by *A* = *LB*(*UB*(*A*)), whereas in general *A* is merely a subset of *LB*(*UB*(*A*)).

### Mappings of ordered sets

A function from an ordered set (*X*,<) to (*Y*,<) is *monotonic* or *monotone increasing* if it preserves order: that is, when *x* and *y* satisfy then . A *monotone decreasing* function similarly reverses the order. A function is *strictly monotonic* if implies : such a function is necessarily injective.

An *order isomorphism*, or simply *isomorphism* between ordered sets is a monotonic bijection.

### Chains

A *chain* is a subset of an ordered set for which the induced order is total. An ordered set satisfies the *ascending chain condition* (ACC) if every strictly increasing chain is finite, and the *descending chain condition* (DCC) if every strictly decreasing chain is finite. An order relation satisfying the DCC is also termed *well-founded*.

A *maximal chain* is a chain which cannot be extended by any element and still be linearly ordered (it is maximal within the family of chains ordered by set-theoretic inclusion).

The *dimension* of an element *x* in an ordered set with **0** is the length *d*(*x*) of a longest maximal chain from **0** to *x*.

### Dilworth's theorem

**Dilworth's theorem** states that the width of an ordered set, the maximal size of an antichain, is equal to the minimal number of chains which together covers the set.

## Lattices

A **lattice** is an ordered set in which any two element set has a supremum and an infimum. We call the supremum the *join* and the infimum the *meet* of the elements *a* and *b*, denoted and respectively.

The join and meet satisfy the properties:

These four properties characterize a lattice. The order relation may be recovered from the join and meet by

### Semi-modular lattices

An **upper semi-modular lattice** satisfies the further property:

- Upper semi-modularity: If then .

Dually, a **lower semi-modular lattice** satisfies

- Lower semi-modularity: If then .

The *Jordan-Dedekind chain condition* holds in a semi-modular (lower or upper) lattice: all finite maximal chains between two given elements have the same length.

### Modular lattices

A **modular lattice** satisfies the further property:

- Modularity: If then

A pair of intervals of the form and are said to be *in perspective*. In a modular lattice, perspective intervals are isomorphic: the maps and are order-isomorphisms.

Modularity implies both forms of semi-modularity and hence the Jordan-Dedekind chain condition. In a modular lattice with **0**, if an element *x* has finite dimension *d*, then all maximal chains from **0** to *x* have the same length *d*.

The dimension is related to the join and meet in a modular lattice by

### Distributive lattices

A **distributive lattice** satisfies the further property:

Distributivity implies modularity for a lattice.

### Complemented lattices

A **complete lattice** is one in which every set has a supremum and an infimum. In particular the lattice must be a bounded order, with bottom and top elements, usually denoted **0** and **1**.

A **complemented lattice** is a lattice with **0** and **1** with the property that for every element *a* there is some element *b* such that and . If the lattice is distributive then the *complement* of *a*, denoted or is unique.

### Subjunctive lattice

A **subjunctive** or **Brouwerian lattice** has the property that for any two elements *a*,*b*, there exists an element *a*→*b* with the properties

This element is the **pseudo-complement** of *a* **relative to** *b* and is unique. We note that *a*→*a* = **1**.

A **Heyting algebra** is a bounded subjunctive lattice. The **pseudo-complement** ~*a* is the relative pseudo-complement *a*→**0**. We have but need not be **1**. A Heyting algebra is necessarily distributive.

### Boolean lattice

A **Boolean lattice** is a distributive complemented lattice, and hence with a uniquely defined complement.

A Boolean lattice is subjunctive.

### Lattice homomorphisms

A **lattice homomorphism** is a map between lattices which preserves join and meet. It is necessarily montone, but not every monotone map is a lattice homomorphism. A lattice isomorphism is just an order isomorphism.

### Ideals and filters

An **ideal** in a lattice is a non-empty join-closed downward-closed subset. A **filter** is a non-empty meet-closed upward-closed subset. Every cut defines an ideal, but not conversely. The downset is the *principal ideal* on *a*; the upset is the *principal filter* on *a*.