Parallel product

The parallel product may refer to one of several closely related concepts. The parallel product is closely related to the mixing of abstract polytopes and the two are generalized by the parallel product on maniplexes.

Groups
The parallel product $$G\|H$$ of two groups $G$ and $H$ generated by the sets $$g_G$$ and $$g_H$$ respectively is a group generated by the set $$g_G \times g_H$$ under the action:

$$ \times_{G\|H} : (G\times H)\times(G\times H)\rightarrow(G\times H) \\ (g, h)\times_{G\|H}(g', h') = (g \times_G g',h\times_G h') $$

This action is the same action as the direct product, and thus $$G\|H\subseteq G\times H$$.

The resulting group depends on the choice of generating sets. Since every group generates itself, the direct product is a special case of the parallel product, although usually a different generating set is chosen.

Examples
Let $$C_n$$ be a cyclic group of order $n$. Such a group always has an element $g$ which generates the entire group. $$C_n\|C_n$$ is then a group generated by $$(g,g)$$ in the direct product group. This group is isomorphic to $$C_n$$. Thus cyclic groups are idempotent.

Actions
If we have some group $G$ and two group actions $$\rho_1 : G \rightarrow \mathrm{Aut}(\Omega_1)$$, and $$\rho_2 : G \rightarrow \mathrm{Aut}(\Omega_2)$$, then we can define the parallel product $$\rho_1\|\rho_2$$ to be an action of $G$ on any orbit of the direct product $$\rho_1\times\rho_2$$. The selection of orbit is akin to the selection of generators in the parallel product on groups. In fact the parallel product on actions is a generalization of the product on groups.

Maniplexes
There are multiple equivalent ways to define maniplexes, thus there are multiple ways to define the parallel product on maniplexes.

Actions
A $n$-maniplex can be defined as a group with a sequence of $n$ generators acting on a set of flags. The parallel product then corresponds to an orbit of the parallel product of the groups with respect to their generators acting on the cartesian product of their flags.

Graphs
For two $n$-colored graphs with vertex sets $$V_1$$ and $$V_2$$, their direct product is an $n$-colored graph with vertex set $$V_1\times V_2$$ such that there is an $i$-edge between two vertices $$(a_1,a_2)$$ and $$(b_1,b_2)$$ iff there is an $i$-edge between $$a_1$$ and $$b_1$$ in the first graph and an $i$-edge between $$a_2$$ and $$b_2$$ in the second graph.

The parallel product of two maniplexes is then a connected component of the direct product of the maniplexes.