Definition:Decreasing/Mapping
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Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping.
Then $\phi$ is decreasing if and only if:
- $\forall x, y \in S: x \preceq_1 y \implies \map \phi y \preceq_2 \map \phi x$
Note that this definition also holds if $S = T$.
Also known as
A decreasing mapping is also known as order-inverting, order-reversing, antitone and non-increasing.
Some sources refer to it as monotonic decreasing or monotone decreasing.
Also see
- Results about decreasing mappings can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.2$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): antitone