Semiringporous

Semiringporous is a term found in some mathematical discussions to describe a porous-like property of subsets within semirings. It is not part of a standard, widely adopted definition, and its precise meaning can vary depending on context and the chosen notion of size or scale in the semiring.

In informal usage, a subset A of a semiring S is described as semiringporous if, at every location in S and at many scales, there are substantial gaps or holes in A that are still compatible with the semiring’s algebraic structure. Formulations typically involve a scale system or a valuation that identifies neighbourhoods around points and measures the “size” of holes relative to those neighbourhoods. A common idea is that for each point and for sufficiently small scales, one can find a hole inside the local region that is not contained in A, with the hole occupying a definite proportion of the region. Because semirings lack a canonical metric in general, these definitions rely on an auxiliary notion of scale, radius, or density tied to the particular semiring under consideration.

The concept is closely related to the classical notion of porosity in metric spaces but adapted to an algebraic setting. Variants may use different ways to quantify holes, and the choice of scale depends on the semiring’s structure, such as in tropical or idempotent semirings, valuation semirings, or other algebraic frameworks.

Applications and discussion remain largely theoretical and exploratory, often aimed at understanding sparsity phenomena in algebraic contexts or guiding constructions in related areas of analysis and combinatorics. See also porosity, semiring, tropical geometry, and valuation theory.