Monotonic

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I made this just to make it blue

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Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it

Definition

A map, $f:X\rightarrow Y$ between two posets, $(X,\sqsubseteq)$ and $(Y,\preceq)$ is monotonic or monotone if:

• $\forall a,b\in X[a\sqsubseteq b\implies f(a)\preceq f(b)]$, or in words:
  • It preserves the ordering.

For a sequence

Recall that a sequence, $({ A_n })_{ n = 1 }^{ \infty }\subseteq X$ (for some poset, $(X,\sqsubseteq)$) can be considered as a mapping:

• $A:\mathbb{N}\rightarrow X$ given by $A:n\mapsto A_n$

We can now apply the above definition directly.

Work needed

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TODO: These

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1. How can we have monotonically decreasing things? Via the dual partial ordering of course! To have $\le$ is to induce a unique $\ge$ - these are distinct orderings.
2. Not sure, but probably some call this isotonic, while monotonic is either increasing or decreasing.
3. Unite with monotonic set function