The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory.

A groupoid is a set G with a unary operation and a partial function Here * is not a binary operation because it is not necessarily defined for all possible pairs of G-elements.

A unary operation f, that is, a map from some set S into itself, is called idempotent if, for all x in S,

A unary operation is idempotent if it maps each element of S to a fixed point of f. For a set with n elements there are

* The defining property of unary idempotence, for x in the domain of f, can equivalently be rewritten as, using the binary operation of function composition denoted by ∘.

Thus, the statement that f is an idempotent unary operation on S is equivalent to the statement that f is an idempotent element with respect to the function composition operation ∘ on functions from S to S.

In mathematical logic and computer science, the Kleene star ( or Kleene operator or Kleene closure ) is a unary operation, either on sets of strings or on sets of symbols or characters.

# REDIRECT unary operation

A natural generalization of the inverse semigroup is to define an ( arbitrary ) unary operation ° such that ( a °)°= a for all a in S ; this endows S with a type < 2, 1 > algebra.

In order to obtain interesting notion ( s ), the unary operation must somehow interact with the semigroup operation.

A 1-ary operation ( or unary operation ) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~ x.

This is inconvenient ; the list of group properties can be simplified to universally quantified equations if we add a nullary operation e and a unary operation ~ in addition to the binary operation *, then list the axioms for these three operations as follows:

If ~ is a unary operation, then h (~ x ) = ~ h ( x ).

* Unary system: S and a single unary operation over S.

* Group: a monoid with a unary operation ( inverse ), giving rise to inverse elements.

We have a language where is a constant symbol and is a unary function and the following axioms:

The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or The signature ( a formal language's non-logical symbols ) for the axioms includes a constant symbol 0 and a unary function symbol S.

For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.

But this is potentially misleading: using a unary input is slower for any given number, not faster ; the distinction is that a binary ( or larger base ) input is proportional to the base 2 ( or larger base ) logarithm of the number while unary input is proportional to the number itself ; so while the run-time and space requirement in unary looks better as function of the input size, it is a worse function of the number that the input represents.

First, the unary * operator applied to a list object inside a function call will expand that list into the arguments of the function call.

In C ++ and PHP, unary prefix & before a formal parameter of a function denotes pass-by-reference.

A Kripke frame F for a propositional relevance language is a triple ( W, R ,*) where W is a set of indices ( or points or worlds ), R is a ternary accessibility relation between indices, and * is a unary function taking indices to indices.

f < sub > n </ sub >( n ) > 2 ↑< sup > n-1 </ sup > n > 2 ↑< sup > n − 2 </ sup > ( n + 3 ) − 3 = A ( n, n ) for n ≥ 2, where A is the Ackermann function ( of which f < sub > ω </ sub > is a unary version ).

He chose the axioms ( see Peano axioms ), in the language of a single unary function symbol S ( short for " successor "), for the set of natural numbers to be:

* Derivation in differential algebra, a unary function satisfying the Leibniz product law

In computer science, a unary operator is a subset of unary function

In the second definition, a function f is called time-constructible if there exists a Turing machine M which, given a string 1 < sup > n </ sup >, outputs the binary representation of f ( n ) in O ( f ( n )) time ( a unary representation may be used instead, since the two can be interconverted in O ( f ( n )) time ).

Equivalently, a function f is space-constructible if there exists a Turing machine M which, given a string 1 < sup > n </ sup > consisting of n ones, outputs the binary ( or unary ) representation of f ( n ), while using only O ( f ( n )) space.

On the other hand, a non-logical predicate symbol such as Phil ( x ) could be interpreted to mean " x is a philosopher ", " x is a man named Philip ", or any other unary predicate, depending on the interpretation at hand.

Also commonly, negation is considered to be a unary connective.

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols.

* Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties.

* A Kleene algebra is an idempotent semiring R with an additional unary operator *: R → R called the Kleene star.

On the other hand, if T is written as a unary number ( a string of n ones, where n = T ), then it only takes time n. By writing T in unary rather than binary, we have reduced the obvious sequential algorithm from exponential time to linear time.

IST is an extension of Zermelo-Fraenkel set theory ( ZF ) in that alongside the basic binary membership relation, it introduces a new unary predicate standard which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

A feature is a concept applied to several fields of linguistics, typically involving the assignment of binary or unary conditions which act as constraints.

The minus sign "−" signifies the operator for both the binary ( two-operand ) operation of subtraction ( as in ) and the unary ( one-operand ) operation of negation ( as in, or twice in ).

Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S ( 0 ), 2 as S ( S ( 0 )) ( which is also S ( 1 )), and, in general, any natural number n as S < sup > n </ sup >( 0 ).

where P is any unary predicate that does not mention A, to define a unique set whose members are precisely the sets satisfying the predicate.

The idea is that κ cannot be distinguished ( looking from below ) from smaller cardinals by any formula of n + 1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol ( for A ).

On the other hand, it also contains some impractical problems, including some undecidable problems such as the unary version of any undecidable problem.

In any numeral system ( except unary, where radix is 1 ), the base is written as " 10 " in that base.