Definition:Increasing/Mapping

Definition

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is increasing iff:

$\forall x, y \in S: x \ \preceq_1 y \ \implies \phi \left({x}\right) \ \preceq_2 \ \phi \left({y}\right)$

Note that this definition also holds if $S = T$.

Also known as

An increasing mapping is also known as order-preserving, isotone and non-decreasing.

Also defined as

Some sources insist at the point of definition that $\phi$ be an injection for it to be definable as order-preserving, but this is conceptually unnecessary.

Also see

• Strictly Increasing Mapping
• Decreasing Mapping
• Monotone Mapping

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$