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 ${\subset },{\supset }$,${\subseteq },{\supseteq }$; ${\prec },{\succ }$,${\preceq },{\succeq }$; ${\sqsubset },{\sqsupset }$,${\sqsubseteq },{\sqsupseteq }$ are also common. We also use the traditional notational convention that $x<y\Leftrightarrow y>x$.

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 (strict) partial order, a relation < on a set satisfying:

• Irreflexive: $x\not <x;\,$
• Antisymmetric: $x<y\Rightarrow y\not <x;\,$
• Transitive: $x<y,y<z\Rightarrow x<z.\,$

The weak form ≤ of an order satisfies the variant conditions:

• Reflexive: $x\leq x;\,$
• Antisymmetric: $x\leq y,y\leq x\Rightarrow x=y;\,$
• Transitive: $x\leq y,y\leq z\Rightarrow x\leq z.\,$

Total order

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

Associated concepts

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

$[a,b]=\{x\in X:a\leq x\leq b\}.\,$

We say that b covers a if the interval $[a,b]=\{a,b\}$: that is, there is no x strictly between a and b. We write $a\prec b$ or $b\succ a$.

Let S be a subset of a ordered set (X,<). An upper bound for S is an element U of X such that $U\geq s$ for all elements $s\in S$. A lower bound for S is an element L of X such that $L\leq s$ for all elements $s\in S$. 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.

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 $\top$ or 1; a minimum element for the whole set may be termed bottom, zero or false and denoted $\bot$ 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

$x\leq s\in S\Rightarrow x\in S.\,$

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

$x\geq s\in S\Rightarrow x\in S.\,$

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 $x\leq y$ then $f(x)\leq f(y)$. A monotone decreasing function similarly reverses the order. A function is strictly monotonic if $x<y$ implies $f(x)<f(y)$: 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 $\{a,b\}$ has a supremum and an infimum. We call the supremum the join and the infimum the meet of the elements a and b, denoted $a\vee b$ and $a\wedge b$ respectively.

The join and meet satisfy the properties:

• Idempotence: $x\vee x=x=x\wedge x;\,$
• Commutativity: $x\vee y=y\vee x;~x\wedge y=y\wedge x;\,$
• Associativity: $x\vee (y\vee z)=(x\vee y)\vee z;~x\wedge (y\wedge z)=(x\wedge y)\wedge z;\,$
• Absorption: $x\wedge (x\vee y)=x;~x\vee (x\wedge y)=x;\,$

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

$a\vee b=b\Leftrightarrow a\leq b\Leftrightarrow a\wedge b=a.\,$

Semi-modular lattices

An upper semi-modular lattice satisfies the further property:

• Upper semi-modularity: If $a\wedge b\prec a$ then $b\prec a\vee b$.

Dually, a lower semi-modular lattice satisfies

• Lower semi-modularity: If $b\prec a\vee b$ then $a\wedge b\prec a$.

Modular lattices

A modular lattice satisfies the further property:

• Modularity: If $x\geq y$ then $x\wedge (y\vee z)=y\vee (x\wedge z).\,$

A pair of intervals of the form $[a\vee b,b]$ and $[a,a\wedge b]$ are said to be in perspective. In a modular lattice, perspective intervals are isomorphic.

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 Jordan-Dedekind chain condition holds in a modular lattice: all finite maximal chains between two given elements have the same length.

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

$d(x)+d(y)=d(x\wedge y)+d(x\vee y).\,$

Distributive lattices

A distributive lattice satisfies the further property:

• Distributivity: $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z);~x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z).\,$

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 have 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 $a\wedge b=\mathbf {0}$ and $a\vee b=\mathbf {1}$. If the lattice is distributive then the complement of a, denoted ${\bar {a}}$ or $a'$ is unique.

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

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.