Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at finitely many points, and $f\prec g$ is defined as $f(n)\leq g(n)$ with finitely many exceptions.

This lattice is large, e.g. you can construct chains and antichains of size continuum, chains of antichains of the same cardinality and so on. Although these constructions are structurally simple, defining them in a precise way gets quite complicated. Therefore I am interested into references dealing with this lattice or sublattices, making the intuitive "largeness" precise.

Background: Y. Barnea and I constructed a set of groups parametrized by elements of $\mathcal{L}$, and want a way to convince group theorists that we really have "many" groups, and don't want to waste a lot of space on constructions which yield results far inferior to what every lattice theorist would immediately see.

Edit: As every function in $\mathcal{L}$ is equivalent to either a function $f$ satisfying $f(0)=0$ or a function $f(n)=n+c$, the interesting part is the sublattice of functions satisfying $f(0)=0$. Also I always assume AC and would not mind too much about assuming CH.