The study introduces a *symmetrization* process that turns an almost symmetric numerical semigroup into a symmetric numerical set.

Almost symmetric semigroups can be represented by Young diagrams. The authors show that these diagrams satisfy a simple numerical identity involving the genus, type, and number of small elements, and a combinatorial hook‑length condition. Using the diagram, they delete specific rows—one for each type less than the total—producing a new diagram that corresponds to a symmetric numerical set. The resulting map, called Φ, sends every almost symmetric semigroup to a symmetric set.

Because the map removes rows, it is not one‑to‑one. Two different almost symmetric semigroups with the same top row in their diagrams can end up in the same symmetric set. The paper illustrates this with the examples S = {0,4,6,8,…} and T = {0,5,6,8,…}, both almost symmetric and mapping to Φ(S) = Φ(T) = {0,3,4,6,…}.

When the authors counted how many objects fall into each category for all Frobenius numbers up to 78, the numbers showed interesting patterns. The ratio |AS_F|/|S_F|—the number of almost symmetric semigroups over the number of symmetric images—grows fast. This means many distinct almost symmetric semigroups collapse onto the same symmetric set. For small Frobenius numbers the ratio starts at 1 and escalates swiftly, reaching values above 2 at F = 78.

Another ratio, |S_F|/|N_{S_F}|, begins at 1 but generally increases, passing the threshold of 2 at F = 78. Here N_{S_F} denotes the subset of symmetric sets that are themselves numerical semigroups. The growth shows that, for larger Fibonacci numbers, a sizable part of the symmetric sets produced by Φ are not closed under addition.

The image of Φ is far from covering all symmetric numerical sets. Many symmetric sets obtained by the procedure are not semigroups, indicating that Φ is not onto. Determining exactly which symmetric sets lie in the image remains an open question.