Nonsimple

Nonsimple is an adjective used to describe objects that are not simple, meaning they possess nontrivial structure, reducible components, or self-intersections. The precise meaning depends on the domain, but the unifying idea is that a nonsimple object cannot be treated as irreducible or elementary.

In group theory, a group is nonsimple if it has nontrivial proper normal subgroups. Simple groups have no such subgroups and are viewed as the building blocks of groups. For example, the symmetric group S3 is nonsimple because it has a normal subgroup A3 of order 3.

In ring theory, a ring is simple if it has no nontrivial two-sided ideals. Many rings are nonsimple; for instance, the ring of integers Z has nontrivial ideals nZ for every positive n.

In graph theory, a simple graph has no loops or multiple edges. A nonsimple graph may contain loops or parallel edges, or otherwise exhibit features that prevent it from being considered simple.

In topology and geometry, a curve is simple if it does not intersect itself; a nonsimple curve or polygon intersects itself, producing self-intersections or crossings.

The term is generally used descriptively; the exact criterion depends on the mathematical structure being discussed. See also simple, simplicity, and nonsimplicity.