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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le y, y \le x \Rightarrow x=y ; \,}
• Transitive: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le y, y \le z \Rightarrow x \le 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < y} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = y} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > y} holds.

Associated concepts

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] = \{ x \in X : a \le x \le b \}.\,}

We say that b covers a if the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] = \{a,b\}} : that is, there is no x strictly between a and b.

Let S be a subset of a ordered set (X,<). An upper bound for S is an element u of X such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \ge s} for all elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in S} . A lower bound for S is an element l of X such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell \ge s} for all elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in S} . In general a set need not have either an upper or a lower 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 a minimum.

Lattices

A lattice is an ordered set in which any two element set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \vee b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \wedge b} respectively.

The join and meet satisfy the properties:

• Idempotence: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \vee x = x = x \wedge x ; \,}
• Commutativity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \vee y = y \vee x;~ x \wedge y = y \wedge x ;\,}
• Associativity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \vee (y \vee z) = (x \vee y) \vee z;~ x \wedge (y \wedge z) = (x \wedge y) \wedge z ;\,}
• Absorption: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \wedge (x \vee y) = x;~ x \vee (x \wedge y) = x;\,}

These four properties characterise a lattice.

A distributive lattice satisfies the further property:

• Distributivity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z);~ x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z ).\,}