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Some structures have a property of closure by a "sum" or "product" operation. Given a family of structures $(S_i)_{i \in I}$, we can then define a new structure denoted by $\sum_{i \in I} S_i$, resp. $\prod_{i \in I} S_i$; structures enjoying this property are vector spaces for instance. I consider these operations as "undirected" as the result does not depend on the order of the summands, up to isomorphism.

I am interested in structures for which we can define a "directed sum" / "directed product" operation, meaning that the definition of the sum / product could now depend on the structure of the index set $I$ (e.g. it could be a totally ordered set and we would compute non-commutative sums / products according to this order).

To illustrate this, consider the following definitions for posets. Let $I = (G,\leq)$ be a poset and for each $i \in G$ let $S_i = (U_i,\leq_i)$ be a poset.

  • we define $\sum_{i \in I} S_i$ as the poset $R = (V,\leq')$ with $V = \cup_{i \in G} \{i\} \times U_i$, and $(i,x) \leq' (j,y)$ iff either $i < j$ or ($i = j$ and $x \leq_i y$);

  • we define $\prod_{i \in I} S_i$ as the poset $T = (W,\leq'')$ with $W = \prod_{i \in G} U_i$, and $x \leq'' y$ iff there exists a maximal antichain $A$ of $I$ such that (i) for every $i \in G$ below $A$, $x_i = y_i$, (ii) for every $i \in A$, $x_i \leq_i y_i$.

Note that when the poset $I$ has no arcs, then we recover the usual constructions (disjoint union and function dominance).

In particular I would like to know whether (i) there could exist an axiomatic / category-theoretic definition of these "directed" operations, (ii) there are other examples of structures displaying this property.

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