Definition:Positive
Contents
Definition
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$.
Then $x \in R$ is positive if and only if $0_R \le x$.
The set of all positive elements of $R$ is denoted:
- $R_{\ge 0_R} := \set {x \in R: 0_R \le x}$
Ordered Integral Domain
Definition 1
An ordered integral domain is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$:
\((\text P 1)\) | $:$ | Closure under Ring Addition: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a + b} \) | ||||
\((\text P 2)\) | $:$ | Closure under Ring Product: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a \times b} \) | ||||
\((\text P 3)\) | $:$ | Trichotomy Law: | \(\displaystyle \forall a \in D:\) | \(\displaystyle \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} \) | ||||
For $\text P 3$, exactly one condition applies for all $a \in D$. |
Definition 2
An ordered integral domain is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain.
That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$:
\((\text {OID} 1)\) | $:$ | $\le$ is compatible with ring addition: | \(\displaystyle \forall a, b, c \in D:\) | \(\displaystyle a \le b \) | \(\displaystyle \implies \) | \(\displaystyle \paren {a + c} \le \paren {b + c} \) | ||
\((\text {OID} 2)\) | $:$ | Strict positivity is closed under ring product: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle 0_D \le a, 0_D \le b \) | \(\displaystyle \implies \) | \(\displaystyle 0_D \le a \times b \) |
An ordered integral domain can be denoted:
- $\struct {D, +, \times \le}$
where $\le$ is the total ordering induced by the strict positivity property.
Numbers
The concept of positive can be applied to the following sets of numbers:
- $(1): \quad$ The integers $\Z$
- $(2): \quad$ The rational numbers $\Q$
- $(3): \quad$ The real numbers $\R$
The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a positive complex number.
As for the natural numbers, they are all positive by dint of their being the non-negative integers.
Integers
Informally, the positive integers are the set:
- $\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$
As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.
Thus positive can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the difference congruence classes, positive can be defined directly as the relation specified as follows:
- The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is positive if and only if $b \le a$.
The set of positive integers is denoted $\Z_{\ge 0}$.
An element of $\Z$ can be specifically indicated as being positive by prepending a $+$ sign:
- $+x := x \in \Z_{\ge 0}$.
Ordering on Integers
Definition 1
The integers are ordered on the relation $\le$ as follows:
- $\forall x, y \in \Z: x \le y$
- $\exists c \in P: x + c = y$
where $P$ is the set of positive integers.
That is, $x$ is less than or equal to $y$ if and only if $y - x$ is non-negative.
Definition 2
The integers are ordered on the relation $\le$ as follows:
Let $x$ and $y$ be defined as from the formal definition of integers:
- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.
Then:
- $x < y \iff x_1 + y_2 \le x_2 + y_1$
where:
- $+$ denotes natural number addition
- $\le$ denotes natural number ordering.
Rational Numbers
The positive rational numbers are the set defined as:
- $\Q_{\ge 0} := \set {x \in \Q: x \ge 0}$
That is, all the rational numbers that are greater than or equal to zero.
Real Numbers
The positive real numbers are the set:
- $\R_{\ge 0} = \set {x \in \R: x \ge 0}$
That is, all the real numbers that are greater than or equal to zero.
Also known as
The notations $R_+$ and $R^+$ are frequently seen for $\set {x \in R: 0_R \le x}$.
However, these notations are also used for $\set {x \in R: 0_R < x}$, that is, $R_{> 0_R}$, and so suffer from being ambiguous.
Also defined as
Some treatments of this subject use the term define non-negative to define $x \in R$ where $0_R \le x$, reserving the term positive for what is defined on this website as strictly positive.
With the conveniently unambiguous notation that has been adopted on this site, the distinction between the terms loses its importance, as the symbology removes the confusion.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$