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## An attempt to generalize universal algebra

Universal constructs has basic relations on the form $\mathcal R^F_X\subseteq F(X)\times X$ for some functor $F:\textbf{Set}\rightarrow\textbf{Rel}$ and has morphisms $f:X\rightarrow Y$ satisfying the relation:

$(1)\quad\; (\phi,\psi)\in F(f)\Rightarrow\left[(\phi,x)\in\mathcal R^F_X\Rightarrow(\psi,f(x))\in\mathcal R^F_Y\right]$.

The normal definition of groups gives an example of an universal construct, but also topological spaces can be defined as universal constructs by:

$(2)\quad\; (A,x)\in\mathcal R^{\mathcal{Q}}_X\Leftrightarrow x\in\textbf{cl}(A)$,

where cl is the Kuratowski closure operator and $\mathcal Q$ is the contravariant powerset functor.

To test $(1)$:

$\varphi:G\rightarrow H$ group homomorphism and $F$ is the product functor $F(\varphi)=\varphi\times\varphi$. Now suppose $((g_1,h_1),(g_2,h_2))\in \varphi\times\varphi$:

$((g_1,g_2),g)\in\mu\Leftrightarrow g=g_1\cdot g_2\Rightarrow$ $h=\varphi(g)=\varphi(g_1\cdot g_2)=\varphi(g_1)\cdot\varphi(g_2)=$ $h_1\cdot h_2=$ $\mu'(\varphi(g_1),\varphi(g_2))=\mu'(h_1,h_2)\Rightarrow$ $((h_1,h_2),h)\in\mu'$

$f:X\rightarrow Y$ continuous function and $(A,B)\in\mathcal Q(f)\Leftrightarrow A=f^{-1}(B)$:

$(A,x)\in \mathcal R^{\mathcal Q}_X\Rightarrow x\in\textbf{cl}(A)=\textbf{cl}(f^{-1}(B)) \Rightarrow f(x)\in \textbf{cl}(B)\Rightarrow$ $(B,f(x))\in \mathcal R^{\mathcal Q}_Y$

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