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Here we present all of the unary and binary operators in the language. They are members of the class Keyword. The binary operators are != # #? % %= & &= * ** **= *= + ++ ++= += , - -= -> . .. ..< ..<= ..= .? / // //= /= : := ; < <- << <<= <= <== <=== <==> <==>= = =!= == === ===> ===>= ==> ==>= => > >= >> >>= ? ?? ??= @ @= @@ @@= \ \= \\ \\= ^ ^** ^**= ^< ^<= ^= ^> ^>= ^^ ^^= _ _< _<= _= _> _>= | |- |-= |= |_ |_= || ||= · (missing documentation) ·= ⊠ (missing documentation) ⊠= ⧢ (missing documentation) ⧢= and or SPACE xor , the unary prefix operators are # * + , - < << <= <== <=== > >= ? ?? |- not , and the unary postfix operators are ! (*) , ; ^! ^* ^~ _! _* _~ ~ .

Of those, the ones for which users may install new methods are the binary operators % %= & &= * ** **= *= + ++ ++= += - -= .. ..< ..<= ..= / // //= /= : << <<= <== <=== <==> <==>= == ===> ===>= ==> ==>= >> >>= ? ?? ??= @ @= @@ @@= \ \= \\ \\= ^ ^** ^**= ^< ^<= ^= ^> ^>= ^^ ^^= _ _< _<= _= _> _>= | |- |-= |= |_ |_= || ||= · (missing documentation) ·= ⊠ (missing documentation) ⊠= ⧢ (missing documentation) ⧢= and or SPACE xor , the unary prefix operators * + - < << <= <== <=== > >= ? ?? |- not , and the unary postfix operators ! (*) ^! ^* ^~ _! _* _~ ~ .

Menu

assignment

• = -- assignment
• := -- assignment of method or new local variable
• <- -- assignment with left side evaluated

equality testing

• == -- equality
• != -- inequality
• === -- strict equality
• =!= -- strict inequality

comparison of objects

• < -- less than
• <= -- less than or equal
• > -- greater than
• >= -- greater than or equal
• ? -- comparison operator

predicates

• and -- conjunction
• or -- disjunction
• xor -- exclusive disjunction
• not -- negation

functions

• SPACE -- blank operator; often used for function application, making polynomial rings
• -> -- make a function
• @@ -- a binary operator
• \\ -- a binary operator
• \ -- a binary operator

subscripting and object access

• _ -- a binary operator, used for subscripting and access to elements
• . -- access to elements whose key is a symbol
• .? -- check for presence of elements whose key is a symbol
• # -- length or access to elements
• #? -- check existence of value in a list, hash table, database, or string

arithmetic operators

• ! -- factorial
• + -- a unary or binary operator, usually used for addition
• - -- a unary or binary operator, usually used for negation or subtraction
• * -- a binary operator, usually used for multiplication
• / -- a binary operator, usually used for division
• // -- a binary operator, usually used for quotient
• % -- a binary operator, usually used for remainder and reduction
• ^ -- a binary operator, usually used for powers
• & -- a binary operator
• ++ -- a binary operator, usually used for direct sum
• ** -- a binary operator, usually used for tensor product or Cartesian product
• ^** -- a binary operator, usually used for tensor or Cartesian power
• ~ -- a unary postfix operator
• (*) -- a unary postfix operator, used for indicating a graded object
• : -- a binary operator, uses include repetition; ideal quotients

miscellaneous operators

• .. -- a binary operator, used for sequences of consecutive items
• ..< -- a binary operator, used for sequences of consecutive items, not including the endpoint
• << -- a binary operator, used for bit shifting or file output
• >> -- a binary operator, used for bit shifting or attaching optional inputs to functions
• | -- a binary operator, often used for horizontal concatenation
• || -- a binary operator, often used for vertical concatenation
• @ -- a binary operator
• ^^ -- logical exclusive-or
• ?? -- null coalescing operator
• |- -- a binary operator
• <==> -- a binary operator
• ===> -- a binary operator
• ==> -- a binary operator
• <=== -- a unary and binary operator
• <== -- a unary and binary operator
• ^< -- a binary operator, used for truncation
• ^<= -- a binary operator, used for truncation
• ^> -- a binary operator, used for truncation
• ^>= -- a binary operator, used for truncation
• _< -- a binary operator, used for truncation
• _<= -- a binary operator, used for truncation
• _> -- a binary operator, used for truncation
• _>= -- a binary operator, used for truncation
• _~ -- a unary postfix operator

operators used for functors

• ^~ -- a unary postfix operator, used for sheafification
• ^! -- a unary postfix operator, used for the upper shriek functor
• ^* -- a unary postfix operator, used for indicating pullback maps
• _! -- a unary postfix operator, used for the lower shriek functor
• _* -- a unary postfix operator, used for indicating pushforward maps
• |_ -- a binary operator, used for restriction to a subset

built in operators

• => -- construct an option
• , -- the comma, used for separating entries in a list or sequence
• ; -- expression separator

augmented assignment

• %= -- augmented assignment for %
• &= -- augmented assignment for &
• **= -- augmented assignment for **
• *= -- augmented assignment for *
• ++= -- augmented assignment for ++
• += -- augmented assignment for +
• -= -- augmented assignment for -
• ..<= -- augmented assignment for ..<
• ..= -- augmented assignment for ..
• //= -- augmented assignment for //
• /= -- augmented assignment for /
• <<= -- augmented assignment for <<
• <==>= -- augmented assignment for <==>
• ===>= -- augmented assignment for ===>
• ==>= -- augmented assignment for ==>
• >>= -- augmented assignment for >>
• ??= -- augmented assignment for ??
• @= -- augmented assignment for @
• @@= -- augmented assignment for @@
• \= -- augmented assignment for \
• \\= -- augmented assignment for \\
• ^**= -- augmented assignment for ^**
• ^= -- augmented assignment for ^
• ^^= -- augmented assignment for ^^
• _= -- augmented assignment for _
• |-= -- augmented assignment for |-
• |= -- augmented assignment for |
• |_= -- augmented assignment for |_
• ||= -- augmented assignment for ||
• ·= -- augmented assignment for ·
• ⊠= -- augmented assignment for ⊠
• ⧢= -- augmented assignment for ⧢

further information

• precedence of operators
• operatorAttributes -- a hash table with information about Macaulay2 operators

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