The Dushnik-Miller dimension of a (discrete) poset, $P$, involves embedding $P$ into $\mathbb{N}^d$, for $d$ minimal. Inherent in the construction is that the embedding preserves the order of $P$. However, there are many poset properties that are not preserved.
For example, let $P$ be the power set of $\{a, b, c\}$ order with inclusion, and let $P$ be a po-semigroup with union being the operation. The dimension is 3, but the embedding $\phi:P\rightarrow \mathbb{N}^3$ does not preserve the semigroup structure. The embedding I used came from linear extensions $$L_1=\{0,a,b,ab,c,ac,bc,abc\} \\L_2=\{0,c,b,bc,a,ac,ab,abc\} \\ L_3=\{0,a,c,ac,b,ab,bc,abc\}$$
$$\phi(a)+\phi(ab)=(1,4,1)+(3,6,5)=(4,10,6)\neq \phi(a\cup ab)=\phi(ab)=(3,6,5)$$The embedding also fails in general to preserve meets and joins if they exist in $P$.
Are there any other obvious or non-obvious properties that are preserved or destroyed by the Dushnik-Miller embedding?