3

A Set under a binary operation forms a Magma.

A Set under a "unary" operation forms a "Function from a Set to itself" (according to Is there a name for a set together with a unary operation?)

What is the name of the Algebraic structures that are formed for a Set under operations which have arity 3 or higher(or unknown arity)?

Also, does it make sense to have an algebraic structure with arity 0?

Community
  • 1
Dmytro
  • 1,155
  • It seems more proper to speak of the arity of an operation, and of an algebraic structure when named operations of specified arity satisfy fixed axioms. – hardmath Sep 23 '16 at 23:42
  • so without axioms of interest, the structure is of no special interest? – Dmytro Sep 23 '16 at 23:44
  • You were correct to point out magma as a set with a binary operation that is not further qualified by properties. Such very abstract settings are not entirely devoid of interest. But in applications the qualifying properties such as associativity generate most of our interest. – hardmath Sep 23 '16 at 23:55
  • if I am not mistaken, magma does require closure. – Dmytro Sep 23 '16 at 23:56
  • 1
    Closure is merely saying that an operation (in this case a binary operation) is defined. This is more a feature of the language of operations, and thus "closure" can be finessed as an axiom. – hardmath Sep 24 '16 at 00:01
  • so it's an axiom that any algebraic structure gets for free?(no such thing as an algebraic structure that does not satisfy closure)? – Dmytro Sep 24 '16 at 00:05
  • When we describe an algebraic language feature as a binary operation, we are imputing to it the character of a function defined for pairs of input arguments. In more general settings we may describe language features as relations or predicates. – hardmath Sep 24 '16 at 00:11
  • @Dmitry an operation is characterized as a function of the kind $$f:A^n\to A$$ Then the operation have arity $n$, but notice that it goes to $A$ so an operation in closed ever. By example: $$+:\Bbb Z^2\to\Bbb Z$$ – Masacroso Sep 24 '16 at 01:01
  • @Masacroso ah so if it's (operator) :: Z -> R, it would not be closed over any particular set since the codomain is not the same set as the domain? Or am I misunderstanding you. – Dmytro Sep 24 '16 at 03:09

1 Answers1

1

An algebraic function with arity zero is a constant symbol. For example, we may describe a ring (with unit) using binary operations $+$, $\cdot$, unary operation $-$, and nullary operations $0, 1$.

GEdgar
  • 111,679
  • For general operations of $n$ (possibly) greater than 3 arguments (three would be called "ternary"), mathematicians and computer scientists usually resort to the neologism "$n$-ary". – hardmath Sep 24 '16 at 16:35