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If you are familiar with this stuff then you probably don't need the information I have added. So let me start with the question:

Can you prove that category $\mathbf{Lat}$ is not a full subcategory of category $\mathbf{Pos}$?

I dived into the theory of universal algebras and encountered lattices. I noticed that a lattice $\left(L,\wedge,\vee\right)$ can be desribed as a poset that is equipped with binary products and coproducts. Here $a\leq b$ is defined by $a=a\wedge b$. Then $a\sqcap b=a\wedge b$ and $a\sqcup b=a\vee b$. There is a category $\mathbf{Lat}$ having lattices as objects and functions that respect $\wedge$ and $\vee$ as arrows. These functions are order-preserving which makes them arrows in $\mathbf{Pos}$ having posets as objects and orderpreserving functions as arrows. So $\mathbf{Lat}$ is a subcategory of $\mathbf{Pos}$ but I 'know' that it is not a full subcategory. As a confirmation of that I am looking for an arrow in $\hom_{\mathbf{Pos}}\left(L,L'\right)$ that is not an arrow in $\hom_{\mathbf{Lat}}\left(L,L'\right)$. Here$L$ and $L'$ are lattices. So it is about a function that is orderpreserving, but does not respect (at least one of) the operations $\wedge,\vee$.

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$$ \begin{matrix} & & D & & & & & & D \\ & \nearrow & & \nwarrow & & & & \nearrow & & \nwarrow \\ B & & & & C & \quad \Longrightarrow \quad & B & & & & C & \\ & \nwarrow & & \nearrow & & & & \nwarrow & & \nearrow \\ & & A & & & & & & E \\ & & & & & & & & \uparrow \\ & & & & & & & & A \end{matrix} $$

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  • $\begingroup$ $A=B\sqcap C$ and $fA\neq fB\sqcap fC=E$ right? Thank you. $\endgroup$ – drhab Jan 13 '14 at 13:38

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