Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-*closed* if the following condition holds:

- for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $f\cdot g\in F$.

Fact: the intersection of two closed subsets of $S$ is closed.

A *coordinate map* for $S$ is a pair $(U,\varphi,G)$ such that $U\subseteq S$ is a closed non-empty subset, $G=(G,\circ)$ is a semicategory, and $\varphi\colon U\rightarrow \mathrm{mor}(G)$ is a bijection such that $(u,v)\in \mathrm{dom}(\cdot)$ iff $(\varphi(u),\varphi(v))\in\circ$ and $(u,v)\in\mathrm{dom}(\cdot)$ implies $\varphi(u\cdot v)=\varphi(u)\circ\varphi(v)$ for all $u,v\in U$.

An *atlas* for $(S,\cdot)$ is a collection of coordinate maps $(U_\alpha,\varphi_\alpha,G_\alpha)$ such that $S = \cup_\alpha U_\alpha$, $\alpha\in I$ an index set.

Fact: the transition maps $\tau_{\alpha,\beta}\colon \varphi_\alpha(U_\alpha \cap U_\beta)\rightarrow \varphi_\beta(U_\alpha \cap U_\beta)$, $\tau_{\alpha,\beta}=\varphi_\beta\circ\varphi_\alpha^{-1}$ are semigroupoid $\circ$-isomorphisms, i.e. a composability- and composition-preserving correspondence of the morphisms.

The pair $(S,\cdot)$ together with an atlas is an *algebraic manifold*.

Has such an "algebraic manifold" been defined before? Are there any non-trivial examples?

3more comments