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Orderless

is an attribute that can be assigned to a symbol f to indicate that the elements ei in expressions of the form f[e1,e2,…] should automatically be sorted into canonical order. This property is accounted for in pattern matching.

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Orderless

Orderless

is an attribute that can be assigned to a symbol f to indicate that the elements ei in expressions of the form f[e1,e2,…] should automatically be sorted into canonical order. This property is accounted for in pattern matching.

Details

  • The Orderless attribute for a function corresponds to the mathematical property of commutativity.
  • Functions with the Orderless attribute use canonical order as described in the notes for Sort.
  • For an object that represents a matrix or a tensor, the Orderless attribute represents symmetry among indices.
  • Functions like Plus and Times are Orderless.
  • In matching patterns with Orderless functions, all possible orders of arguments are tried.
  • The Orderless attribute must be assigned before defining any values for an Orderless function.

Examples

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Basic Examples  (3)

Plus sorts its arguments because it has attribute Orderless:

Wolfram Language code: b + a
Wolfram Language code: Attributes[Plus]

Any orderless function will sort its arguments:

Wolfram Language code: SetAttributes[f, Orderless];
Wolfram Language code: f[b, a]

The pattern matcher reorders arguments of Orderless functions to obtain matches:

Wolfram Language code: MatchQ[2 + x, x + _Integer]

Note that Plus itself does not reorder these particular arguments:

Wolfram Language code: 2 + x
Wolfram Language code: x + _Integer

Properties & Relations  (2)

All possible argument orderings are tried to match a definition for an orderless function:

Wolfram Language code: SetAttributes[f, Orderless]; f[x_Integer, y_Real] := y ^ x
Wolfram Language code: f[2, 3.0]
Wolfram Language code: f[3.0, 2]

Elements of an expression whose head is orderless are sorted into standard order:

Wolfram Language code: SetAttributes[h, Orderless];
Wolfram Language code: h[z, -1.0, g[], I, 1]
Wolfram Language code: OrderedQ[%]

This sorting occurs as part of the standard evaluation process:

Wolfram Language code: Hold[h[b, a]]
Wolfram Language code: %//ReleaseHold

Possible Issues  (1)

Pattern matching with orderless functions can lead to a large number of possible cases:

Wolfram Language code: SetAttributes[h, Orderless];
Wolfram Language code: ReplaceList[h[a, b, c], h[x_, y_, z_] :> {x, y, z}]
Wolfram Research (1988), Orderless, Wolfram Language function, https://reference.wolfram.com/language/ref/Orderless.html.

Text

Wolfram Research (1988), Orderless, Wolfram Language function, https://reference.wolfram.com/language/ref/Orderless.html.

CMS

Wolfram Language. 1988. "Orderless." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Orderless.html.

APA

Wolfram Language. (1988). Orderless. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Orderless.html

BibTeX

@misc{reference.wolfram_2026_orderless, author="Wolfram Research", title="{Orderless}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Orderless.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_orderless, organization={Wolfram Research}, title={Orderless}, year={1988}, url={https://reference.wolfram.com/language/ref/Orderless.html}, note=[Accessed: 13-June-2026]}