mikroobsets

Mikroobsets are a mathematical framework for encoding local order information within a set. A mikroobset consists of a set X together with a collection F of finite subsets of X, called microcontexts. To each C in F there is a binary relation ≤_C on C that is a total order. The family {≤_C}_{C∈F} is coherent: if C ⊆ D with C, D ∈ F, then the restriction of ≤_D to C equals ≤_C. Thus elements that appear together in a microcontext have a fixed order, while pairs that never share a microcontext are not ordered by the mikroobset.

Microcontexts may overlap; coherence enforces consistent orderings on overlaps. Mikroobsets generalize ordinary posets by allowing a global order to be replaced with local orders. A simple special case is a global order obtained by taking F = {X} and ≤_X as the usual order; another is a collection of disjoint microcontexts imposing independent local orders.

Construction and morphisms: submikroobsets arise by restricting X and the relevant microcontexts. Products combine two mikroobsets by Cartesian product and intersecting microcontexts. A morphism between mikroobsets is a function f: X → Y that preserves all microcontext orders: for every C ∈ F there is a D ∈ G with f[C] ⊆ D and ≤_D restricted to f[C] agrees with ≤_C.

Applications and notes: mikroobsets model systems with local ordering when a global order is unavailable, such as locally ordered networks or distributed decision processes. The notion also features in discrete geometry and data analysis as a way to describe microstructures within larger datasets.

See also related concepts include partially ordered sets, hypergraphs, and simplicial complexes.