Operations for two-colored partitions

The set $\Pscr^{\circ\bullet}$ of two-colored partitions forms a strict involutive monoidal category via the operations of composition, tensor product and involution.

Composition

Given two-colored partitions $p,p'\in\Pscr^{\circ\bullet}$ the pairing $(p,p')$ is called composable if the upper row of $p$ and the lower row of $p'$ agree in size and coloring.

If $(p,p')$ is composable, if for every $x\in \{p,p'\}$ we denote the lower row of $x$ by $R_{L,x}$ and the upper row by $R_{U,x}$ , if we identify $R_{U,p}$ and $R_{L,p'}$ with each other and if

$s=(\{B\backslash R_{L,p}\,\vert\, B\text{ block of } p\}\backslash\{\emptyset\})\vee (\{B'\backslash R_{U,p'}\,\vert\, B'\text{ block of } p'\}\backslash\{\emptyset\}),$

is the join of the partitions induced by $p$ on $R_{U,p}$ and by $p'$ on $R_{L,p'}$ , then the composition of $(p,p')$ is given by the two-colored partition $pp'$ with lower row $R_{L,p}$ , with upper row $R_{U,p'}$ , with blocks

$\left\{\left(\bigcup_{\substack{B\text{ block of }p\\B\cap D\neq \emptyset}}(R_{L,p}\cap B)\right)\,\cup\,\left(\bigcup_{\substack{B'\text{ block of }p'\\B'\cap D\neq \emptyset}}(R_{U,p'}\cap B')\right)\,\bigg\vert\, D\text{ block of }s\right\}\backslash\{\emptyset\},$

and with the coloring of $p$ on $R_{L,p}$ and that of $p'$ on $R_{U,p'}$ .

Tensor Product

If $p_1,p_2\Pscr^{\circ\bullet}$ are two-colored partitions, if for every $i\in\{1,2\}$ we denote the lower row and its total order of $p_i$ by $(R_{L,p_i},\leq_{L,p_i})$ and, likewise, the upper row by $(R_{U,p_i},\leq_{U,p_i})$ and if we denote the coloring of $p_i$ by $c_{p_i}:R_{L,p_i}\cup R_{U,p_i}\to \{\circ,\bullet\}$ , then tensor product of $(p_1,p_2)$ is given by the two-colored partition satisfying the following conditions:

The set of lower points of $p_1\otimes p_2$ is given by a disjoint union $R_{L,p_1}\sqcup R_{L,p_2}$ of $R_{L,p_1}$ and $R_{L,p_2}$ and, likewise, the set of upper points is a disjoint union $R_{U,p_1}\sqcup R_{U,p_2}$ .
For every row $X\in \{L,U\}$ the row $R_{X,p_1}\sqcup R_{X,p_2}$ of $p_1\otimes p_2$ is equipped with the total order which for every $i\in\{1,2\}$ restricts to $\leq_{X,p_i}$ on $R_{X,p_i}$ and which satisfies $j_1\leq j_2$ for every $j_1\in R_{X,p_1}$ and $j_2\in R_{X,p_2}$ .
The coloring of $p_1\otimes p_2$ is the mapping $(R_{L,p_1}\sqcup R_{L,p_2})\cup(R_{U,p_1}\sqcup R_{U,p_2})\to\{\circ,\bullet\}$ which for every $i\in \{1,2\}$ restrict to $c_{p_i}$ on $R_{L,p_i}\cup R_{U,p_i}$ .
The set of blocks of $p_1\otimes p_2$ is given by the disjoint union $\{ \text{blocks of }p_1\}\sqcup \{\text{blocks of }p_2\}$ , where for every $i\in\{1,2\}$ we interpret the blocks of $p_i$ as subsets not of $R_{L,p_i}\cup R_{U,p_i}$ but of $(R_{L,p_1}\sqcup R_{L,p_2})\cup (R_{U,p_1}\sqcup R_{U,p_2})$ .

Involution

For any two-colored partition $p\in\Pscr$ , if we denote by $(R_L,\leq_L)$ the lower row and by $(R_U,\leq_L)$ the upper row of $p$ , by $\pi$ the blocks of $p$ and by $c:R_L\sqcup R_U\to\{\circ,\bullet\}$ the coloring of $p$ , then the involution of $p$ is given by the partition $p^\ast$ whose lower row is $(R_U,\leq_U)$ , whose upper row is $(R_L,\leq_L)$ , whose blocks are $\pi$ and whose coloring is $c$ .

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Operations for two-colored partitions

Composition

Tensor Product

Involution

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