‣ FullPointsOfUnitalsBlocks( u, b1, b2 ) | ( operation ) |
Returns: The list full point of u w.r.t. the blocks b1,b2. The arguments b1,b2 are either blocks of the unital u, or indices of blocks in BlocksOfUnital( u ).
According to the paper [KSS18] by Korchmáros, Siciliano and Szőnyi, the point \(P\) is a full point of the unital \(U\) w.r.t. the blocks \(b_1,b_2\) if \(P\) is not contained in \(b_1\) or \(b_2\), and, the projection with center \(P\) from \(b_1\) to \(b_2\) is a well-defined bijection.
‣ FullPointsOfUnitalRepresentatives( u ) | ( attribute ) |
Returns: A list of records r containing the fields r.block1, r.block2, r.fullpts, where r.fullpts is the set of full point of u w.r.t. the blocks r.block1, r.block2. The returned list contains all possible full points of u up to the automorphism group of u. That is, if \(P\) is a full point w.r.t. the blocks \(b_1,b_2\), then there is an automorphism \(\alpha\) of \(U\) such that \(P^\alpha, b_1^\alpha, b_2^\alpha\) are in the list.
‣ PerspectivityGroupOfUnitalsBlocks( u, b1, b2[, fullpts] ) | ( operation ) |
Returns: The group generated by perspectivies from block b1 to block b2 of the unital u. Notice that the returned group consists of permutations of [1..Order(u)+1]. A list of full points can be given as 4th argument. It is not checked if the elements of fullpts are full points.
Perspectivities between blocks \(b_1, b_2\) of an abstract unital \(U\) are projections from \(b_1\) to \(b_2\) from a center \(P\). In order the perspectivity be well-defined, \(P\) must be a full point w.r.t. \(b_1, b_2\).
‣ EmbeddedDual3NetsOfUnitalRepresentatives( u ) | ( attribute ) |
Returns: A list of lists each having the form [ b1, b2, b3 ], where b1, b2, b3 are three blocks of the unital u forming an embedded dual 3-net. The returned list contains all possible embedded dual 3-nets of u up to the automorphism group of u. That is, if the blocks \(b_1,b_2,b_3\) form an embedded dual 3-net, then there is an automorphism \(\alpha\) of \(U\) such that \(b_1^\alpha, b_2^\alpha, b_3^\alpha\) are in the list.
‣ LatinSquareOfEmbeddedDual3Net( u, ed3net ) | ( operation ) |
Returns: A latin square associated to the embedded dual 3-net ed3net of the unital u.
‣ IsFullPointRegularUnital( u ) | ( property ) |
Returns: The boolean true if the unital u is full point regular, false otherwise.
The unital \(U\) is said to be full point regular, if for each non-intersecting pair of blocks \(b_1, b_2\) the triple \((U, b_1, b_2)\) is full point regular.
‣ IsStronglyFullPointRegularUnital( u ) | ( property ) |
Returns: The boolean true if the unital u is strongly full point regular, false otherwise.
The unital \(U\) is said to be strongly full point regular, if for each non-intersecting pair of blocks \(b_1, b_2\) the triple \((U, b_1, b_2)\) is full point regular.
generated by GAPDoc2HTML