regular map

plural regular maps

1. algebraic geometryA morphism between algebraic varieties.

   We now find out how a regular map acts on the ring of regular functions on a closed set.[…]Moreover, regular maps can be characterised as the maps that take regular functions into regular functions.

   1994, Igor R. Shafarevich, translated by Miles Reid, Basic Algebraic Geometry 1, 2nd edition, Springer, page 29

   2 more examples

   Since one expects that the canonical projection f#58;X#92;rightarrowX#47;G is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of X.

   2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii

   2017, José F. Fernando, José M. Gamboa, Carlos Ueno, Polynomial, regular and Nash images of Euclidean spaces, Fabrizio Broglia, Françoise Delon, Max Dickmann, Danielle Gondard-Cozette, Victoria Ann Powers (editors), Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 160, The 1-dimensional semialgebraic set 𝒮:=x>0,xy=1 is the image of the regular map f: R ²→ R ²,(x,y)↦((xy-1)²+x²,1/((xy-1)²+x²)).

   Synonyms: morphism

2. graph theoryA symmetric tessellation of a closed surface; a decomposition of a two-dimensional manifold into topological disks such that every flag (incident vertex-edge-face triple) can be transformed into any other flag by a symmetry (i.e., an automorphism) of the decomposition.

   1990 [McGraw-Hill], Jay Kappraff, Connections: The Geometric Bridge Between Art and Science, 2001, World Scientific, page 141, Just as there are only five regular maps on the sphere (or plane), there are only three classes of regular maps that can be created on a torus.

   2 more examples

   If n is finite, the regular map is a tessellation of the plane by congruent n-sided polygons, m of which meet at each vertex.

   2010, Jozef Širáñ, Yan Wang, “Maps with highest level of symmetry that are even more symmetric than other such maps: Regular maps with largest exponent groups”, in Richard A. Brualdi, Samat Hedayat, Hadi Kharaghani, Gholamreza B. Khosrovshahi, Shahriar Shahriari, editors, Combinatorics and Graphs: The Twentieth Anniversary Conference of IPM Combinatorics, American Mathematical Society, page 98

   If M is a regular map of type #92;#123;p,q#92;#125;, then #92;mathit#123;Aut#125;(M)#92;approx#92;Delta(p,q,2)#47;N for some normal subgroup N#92;#92;trianglelefteq#92;#92;Delta(p,q,2). Similar statements hold for the class of orientably regular maps and subgroups of #92;Delta#43; and #92;Delta#43;(p,q,2).

   2013, Roman Nedela, Martin Škoviera, “7.6: Maps”, in Jonathan L. Gross, Jay Yellen, Ping Zhang, editors, Handbook of Graph Theory, 2nd edition, CRC Press, page 845