‣ IsAbstractUnitalDesign( object ) | ( filter ) |
Returns: true or false
A GAP category of abstract unitals. Synonym is IsAbstractUnital.
‣ AbstractUnitalByBlistList( bmat ) | ( function ) |
Returns: The unital object corresponding to the blist list bmat.
Each row of bmat corresponds to a block of the unital. We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong bmat matrix size drops error. The function stores bmat and sets the Order of the unital.
‣ AbstractUnitalByDesignBlocks( blocklist ) | ( function ) |
Returns: The unital object corresponding to the list of blocks blocklist. We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong number of blocks or wrong number of points (union of the blocks in blocklist) drops error. The function stores bmat, which is based on blocklist, sets the Order of the unital and sets the names of the points, PointNamesOfUnital of the unital.
‣ AbstractUnitalByIncidenceMatrix( incmat ) | ( function ) |
Returns: The unital object corresponding to the incidence matrix incmat.
Each row of incmat corresponds to a block of the unital. We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong incmat matrix size drops error. The function stores bmat, which is based on incmat and sets the Order of the unital.
‣ InfoUnitalSZ | ( info class ) |
An infoclass for the package. Its default value is \(0\).
‣ PointsOfUnital( u ) | ( attribute ) |
Returns: The range [ 1..q^3 + 1 ].
If u is a unital of order \(q\), then u has \(q^3 + 1\) points.
‣ BlocksOfUnital( u ) | ( attribute ) |
Returns: The blocks of the unital u.
If u is a unital of order \(q\), then each block is a subset of the points of the unital with \(q + 1\) points. The blocks of an abstract unital form a \(2-(q^3+1,q+1,1)\) design.
‣ PointNamesOfUnital( u ) | ( attribute ) |
Returns: The names of the \(q^3+1\) points of u.
The names of the points of u is a list of length \(q^3+1\) of arbitrary GAP objects. It may be set by SetPointNamesOfUnital. The default is the range [ 1..q^3 + 1 ].
‣ IncidenceDigraph( u ) | ( attribute ) |
Returns: The (bipartite) digraph constructed from the boolean incidence matrix bmat of the unital u.
‣ AutomorphismGroup( u ) | ( attribute ) |
Returns: The automorphism group of the unital u.
The function computes the automorphism group of u with the help of its incidence digraph.
‣ Isomorphism( u1, u2 ) | ( operation ) |
Returns: An isomorphism between the unitals u1 and u1 if they are isomorphic, and fail otherwise.
The isomorphism is a permutation which sends the points of the unital u1 to the points of the unital u2 such that the it preserves the incidence between the points and the blocks. The function computes the isomorphism with the help of the incidence digraphs of the unitals u1 and u2.
‣ UnitalBlistList_axiomcheck@( bmat ) | ( function ) |
Returns: true if bmat is the blist list of an abstract unital.
Each row of bmat corresponds to a block of the unital. We check the sizes of the blocks and the sizes of the intersections of the dual blocks.
‣ IsUnitalBlistList@( bmat ) | ( function ) |
Returns: true if bmat is the blist list of an abstract unital.
Each row of bmat corresponds to a block of the unital. We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong bmat matrix size drops error.
‣ IsUnitalIncidenceMatrix@( incmat ) | ( function ) |
Returns: true if incmat is the incidence matrix of an abstract unital.
Each row of incmat corresponds to a block of the unital. We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong incmat matrix size drops error.
‣ IsUnitalBlockDesign@( blocklist ) | ( function ) |
Returns: true if blocklist is the list of blocks of an abstract unital.
We check the sizes of the blocks and the sizes of the intersections of the dual blocks. Wrong number of blocks or wrong number of points (union of the blocks in blocklist) drops error.
‣ UnitalByBlistListNC@( bmat ) | ( function ) |
Returns: The unital object corresponding to the blist list bmat.
The function stores bmat and sets the order of the unital. The function do not check the necessary conditions (the size of bmat, the sizes of the blocks and their intersections).
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