Semireguläre

Semireguläre is the German term for semiregular, used in mathematics to denote a relaxed form of regularity that appears in several areas such as group actions, tilings, and topology. The common idea is uniformity or symmetry that is not complete or global.

In permutation group theory, a group action of a group G on a set X is called semiregular if the stabilizer of every point is trivial; that is, the only group element fixing a point is the identity. If the action is also transitive, it is called regular, and then every orbit has size equal to the order of the group. Semiregular actions illustrate how symmetry can be present without fixing individual points, and they underpin constructions where the group acts freely on the set.

In geometry, semireguläre tilings (often called Archimedean tilings in English) are tilings of the plane by regular polygons in which all vertex configurations are identical, so the tiling is vertex-transitive but not necessarily symmetric with respect to all elements of the tiling. Classic examples include tilings by triangles and hexagons, and configurations like 3.12.12 or 4.8.8, which balance regular polygon faces with uniform vertex arrangements.

In topology, a space is semireguläre if the regular open sets form a base for the topology. A regular open set is an open set equal to the interior of its closure. When these regular open sets form a base, the space is called semiregulär, providing a middle ground between general topological spaces and spaces with stronger separation properties.

Semireguläre describes several related but distinct notions of partial regularity, unified by the principle of structured uniformity without full regularity.