Sorry for all the mistakes in the original! I think they're mostly fixed now.
Thank you for your patience.
Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, the usual definition of homomorphism is a function $f : A \rightarrow B$ that furthermore satisfies the following requirements.
$\quad$ Condition I. $a \leq_A a' \rightarrow f(a) \leq_B f(a')$
$\quad$ Condition IIa. $f(O_A(a_0,\cdots,a_{n-1}))=O_B(f(a_0),\cdots,f(a_{n-1}))$
Of course, IIa should be taken to hold for all natural $n$ and all function symbols $O$ of $\sigma$ having airity $n$.
Okay, so those are the usual definitions; however, there is a very natural generalization whereby IIa is weakened to the following.
$\quad$ Condition IIb. $f(O_A(a_0,\cdots,a_{n-1})) \leq_B O_B(f(a_0),\cdots,f(a_{n-1}))$
Lets call functions satisfying both I and IIb subhomomorphisms.
Example. Let $A$ denote the ordered monoid (written additively), with underlying set $\{0,1,2,\cdots\}$ and the usual notion of addition, but the reverse ordering.
Then the unique function $f : A \rightarrow A$ with defining property $f(a)=a^2$ is a subhomomorphism, since we can argue as follows.
$$f(a+b) = a^2+b^2 + 2ab \geq a^2+b^2 = f(a)+f(b)$$
Question 1. Do subhomomorphisms have a standard name, and where can I learn more about them?
Note that, as an immediate consequence of the definitions, the identity function is always a subhomomorphism, and the composition of two subhomomorphisms is also a subhomomorphism. Thus, classes of ordered algebraic structures over a common signature form naturally into categories whose arrows are precisely the subhomomorphisms.
Actually, they form naturally into 2-posets, where by "2-poset" I just mean a category enriched in $\mathrm{Pos}$, which somewhat motivates my next question.
Part 2. Given a pair of 2-posets $\mathcal{C}$ and $\mathcal{D}$, there is an obvious definition of "subfunctor" $F : \mathcal{C} \rightarrow \mathcal{D},$ obtained by weakening the usual demands on $F.$ In particular, we're merely assuming the following.
- $F(\mathrm{id}_X) \leq \mathrm{id}_{F(X)},$ for all objects $X$ of $\mathcal{C}$.
- $F(g \circ f) \leq F(g) \circ F(f),$ for all serial pairs of arrows $f,g$ of $\mathcal{C}$.
We must also add the stipulation that $F$ preserves order when restricted to any given homset; otherwise, the composition of a pair of compatible subfunctors needn't be a subfunctor.
Question 2. Do subfunctors between 2-posets have a standard name, and where can I learn more about them?
Motivation. I think that, if $\mathcal{C}$ and $\mathcal{D}$ are 2-posets whose objects are ordered algebraic structures and whose arrows are subhomomorphisms, then it would be most natural to view $\mathrm{Hom}(\mathcal{C},\mathcal{D})$ as consisting of all subfunctors. I have yet to find any evidence for this hunch.
Note that many new phenomena emerge in the passage from functors to subfunctors. In particular, while the value assigned by a functor to the hypotenuse of a commutative triangle is determined by the values it assigns to the other arrows, this is not true for an arbitrary subfunctor.