Makeisets

Makeisets are a class of mathematical objects defined as sets generated by an operation called the make operation applied to a starting seed set. The term combines make and sets, highlighting the constructive nature of these objects. In a typical formulation, a makeiset M arises from an initial seed S0 contained in a fixed universe U by repeatedly applying a rule function f: P(U) → P(U), producing a sequence Sn+1 = f(Sn). The collection of all sets generated in this way, together with possible unions or intersections under a fixed collection of set operations, forms the makeiset family.

Construction and variation: A common approach uses a monotone, inflationary make operation, which ensures the process builds larger sets and often converges to a closed family. Different choices of the rule f lead to diverse families of makeisets; some may stabilize after finitely many steps, while others continue to grow indefinitely. The resulting structure can exhibit lattice-like behavior under inclusion, especially when the rule respects union and intersection.

Examples: If U is the natural numbers and the seed is S0 = {0}, a rule f(S) = S ∪ {max(S) + 1} yields the ascending chain {0}, {0,1}, {0,1,2}, … which forms a simple makeiset family. Another example uses f(S) = S ∪ {s + 1 : s ∈ S}, creating a closed under successor set system.

Applications: Makeisets serve as a conceptual tool for illustrating iterative closure processes in set theory, combinatorics, and computer science. They are often used in teaching or as a framework for exploring how simple rules generate complex families of sets.

See also: set, closure, lattice, generators, iterative processes.