Let

$X$ be a completely regular space,

$C\left(X\right)$ the set of all continuous functions from

$X$ into the real numbers

$\mathbb{R}$, and

$L\subset C\left(X\right)$ a unital vector lattice. The structural space

$H\left(L\right)$ is the set of all lattice homomorphisms from

$L$ into

$\mathbb{R}$. For each

$x\in X$ the point evaluation map

${\delta}_{x}$ belongs to

$H\left(L\right)$, where

${\delta}_{x}\left(f\right)=f\left(x\right)$ for each

$f\in L$. If

$L$ separates points and closed sets, then

$X$ is a topolopical subspace of

$H\left(L\right)$, and

$H\left(L\right)$ is a realcompactification of

$X$. Indeed, all realcompactifications of

$X$ have the form

$H\left(L\right)$ for some

$L\subset C\left(X\right)$. In the present paper, the authors study the realcompactifications

$H\left(L\right)$ and compactifications

$H\left({L}^{*}\right)$ of

$X$, where

${L}^{*}$ denotes the set of bounded elements in

$L$, and

$H\left({C}^{*}\left(X\right)\right)$ is the Stone-Čech compactification

$\beta X$ of

$X$. They call

$X$ $L$-realcompact in case

$H\left(L\right)=X$. They show, among other things, that for each

$\varphi \in H\left(L\right)$ there exists

$\xi \in \beta X$ such that, for all

$f\in L$,

${f}^{\beta}\left(\xi \right)\ne \infty $ and

$\varphi \left(f\right)={f}^{\beta}\left(\xi \right)$. They study the case where

$L=\text{Lip}\left(X\right)$ is the set of real valued Lipschitz functions on a metric space

$X$ and show that

$X$ is

$L$-realcompact if, and only if, every closed ball in

$X$ is compact. Other results concern a vector lattice isomorphism

$T$ between

$\text{Lip}\left(X\right)$ and

$\text{Lip}\left(Y\right)$ which maps unit element to unit element. They show that if

$X,Y$ are complete, then

$T$ exists if, and only if,

$X$ and

$Y$ are Lipschitz homeomorphic. A similar result for

${\text{Lip}}^{*}\left(X\right)$ and

${\text{Lip}}^{*}\left(Y\right)$ was obtained by

*N. Weaver* [Pac. J. Math. 164, No. 1, 179–193 (1994;

Zbl 0797.46007)].