trait
OrderedField extends OrderedRing with Field
Abstract Value Members
-
abstract
def
unit: Element
-
abstract
def
zero: Element
Concrete Value Members
-
final
def
!=(arg0: Any): Boolean
-
final
def
##(): Int
-
final
def
==(arg0: Any): Boolean
-
final
def
asInstanceOf[T0]: T0
-
def
clone(): AnyRef
-
final
def
eq(arg0: AnyRef): Boolean
-
def
equals(arg0: Any): Boolean
-
def
finalize(): Unit
-
final
def
getClass(): Class[_]
-
def
hashCode(): Int
-
final
def
isInstanceOf[T0]: Boolean
-
final
def
ne(arg0: AnyRef): Boolean
-
final
def
notify(): Unit
-
final
def
notifyAll(): Unit
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
-
def
toString(): String
-
final
def
wait(): Unit
-
final
def
wait(arg0: Long, arg1: Int): Unit
-
final
def
wait(arg0: Long): Unit
Inherited from AnyRef
Inherited from Any
A totally ordered abstract field structure. Addition associates and commutes, and multiplication associates, commutes, and distributes over addition. Addition and multiplication both have an identity element, every every element has an additive inverse, and every element except zero has a multiplicative inverse. To the extent practicable, the following axioms should hold.
Axioms for addition:
this
, then their sum 𝑎 + 𝑏 is also an element inthis
.this
.this
.this
has an elementzero
such thatzero
+ 𝑎 == 𝑎 for every element 𝑎 inthis
.this
corresponds an element -𝑎 inthis
such that 𝑎 + (-𝑎) ==zero
.Axioms for multiplication:
this
, then their product 𝑎 * 𝑏 is also an element inthis
.this
.this
.this
has an elementunit
!=zero
such thatunit
* 𝑎 == 𝑎 for every element 𝑎 inthis
.this
and 𝑎 !=zero
then there exists an element 𝑎.inverse
such that 𝑎 * 𝑎.inverse
==unit
.The distributive law:
this
.Order axioms:
this
.this
.this
.0.1
0.0