Franklin's Notes

Morphisms between linear orders

The following is a list of some different types of useful morphisms between linear orderings, distinguished by how they interact with the ordering.

• A monotone function is a function $f:A\to B$ is a function between linear orders such that either $x\leq y\implies f(x)\leq f(y)$ for all $x,y\in A$, or $x\leq y\implies f(x)\geq f(y)$ for all $x,y\in A$.

• An order-preserving function or monotone increasing function, also called a (order) homomorphism, is a morphism in the category $\mathsf{LinOrd}$, or a function $f:A\to B$ between linear orders such that $x\leq y$ implies $f(x)\leq f(y)$.

• An order-embedding or strictly increasing function is a function $f:A\to B$ between linear orders such that $x<y$ implies $f(x)<f(y)$. This can also be characterized as an injective order-preserving function, or a monomorphism in $\mathsf{LinOrd}$.

• A rigid order-preserving function is a function $f:A\to B$ between linear orders with the intermediate value property , so that $f(a)\leq b\leq f(a')$ implies the existence of $a''\in A$ such that $f(a'')=b$. This can also be characterized as an order-preserving function that preserves intervals .

• A successor-preserving function is an order-preserving function $f:A\to B$ such that $f(x')$ is the successor of $f(x)$ in $B$ whenever $x'$ is the successor of $x$ in $A$. Every rigid order-embedding is successor-preserving.

• An order-isomorphism is a bijective order-preserving function. An automorphism is an order-isomorphism of an ordered set with itself.