Nonsubdividable

Nonsubdividable is a term used informally in mathematics and related fields to describe an object that cannot be subdivided into smaller parts under a specified subdivision rule without violating a prescribed property. It is not a standardized technical term with a single universal definition; its exact meaning depends on the context and the rules chosen for subdivision.

In number theory, a common interpretation aligns with irreducibility: a positive integer greater than 1 is nonsubdividable under multiplication if it cannot be expressed as a product of two smaller integers greater than 1. In this sense, primes are examples of nonsubdividable numbers. In ring theory, an irreducible element plays a similar role, being not expressible as a product of smaller nonunit elements.

In geometry or combinatorics, a nonsubdividable object may refer to a shape or structure that cannot be partitioned into a prescribed number of smaller pieces that preserve certain invariants, such as congruence, area, or other properties. For example, under a fixed subdivision rule, a region might be called nonsubdividable if any nontrivial subdivision produces parts that fail to meet the rule.

In lattice theory or module theory, the concept parallels indecomposable or atomic objects, which cannot be decomposed into a direct sum of smaller nonzero objects under the given operation.

Because the property is defined relative to the subdivision rule, the term is inherently context-dependent. See also irreducible, prime element, indecomposable, atomic, subdivision.