In incidence structures, particularly simple ones, blocks are
basically sets. For this reason, the elementary set operations
such as join, meet and subset have been made to work
on blocks. However, blocks are not true Magma enumerated sets, and
so the functions Set and Support below have been provided to
convert a block to an enumerated set of points for other uses.
p in B : IncPt, IncBlk -> BoolElt
True if point p lies in block B, otherwise false.
True if point p does not lie in block B, otherwise false.
Given a subset S of the point set of the incidence structure D and a block B of D, return true if the subset S of points lies in B, otherwise false.
Given a subset S of the point set of the incidence structure D and a block B of D, return true if the subset S of points does not lie in B, otherwise false.
The number of blocks of D that contain the point p.
The number of points contained in the block B of D.
The set of points contained in the block B.
The set of underlying points contained in the block B (i.e. the elements of the set have their "real" types; they are no longer from the category IncPt).
True iff the set (or block) S represents a block of the incidence structure D. If true, also returns one such block.
A block of D containing the points p and q (if one exists). In linear spaces, such a block exists and is unique (assuming p and q are different).
True iff there exists a parallel class of D containing the blocks B and C; if true, also returns one such class.
The connection number c(p, B); i.e. the number of blocks joining p to B in D.
> D, P, B := Design< 2, 7 | {3, 5, 6, 7}, {2, 4, 5, 6}, {1, 4, 6, 7}, > {2, 3, 4, 7}, {1, 2, 5, 7}, {1, 2, 3, 6}, {1, 3, 4, 5} >; > D: Maximal; 2-(7, 4, 2) Design with 7 blocks Points: {@ 1, 2, 3, 4, 5, 6, 7 @} Blocks: {3, 5, 6, 7}, {2, 4, 5, 6}, {1, 4, 6, 7}, {2, 3, 4, 7}, {1, 2, 5, 7}, {1, 2, 3, 6}, {1, 3, 4, 5} > P.1 in B.1; false > P.1 in B.3; true > {P| 1, 2} subset B.5; true > Block(D, P.1, P.2); {1, 2, 5, 7} > b := B.4; > b; {2, 3, 4, 7} > b meet {2, 8}; { 2 } > S := Set(b); > S, Universe(S); { 2, 3, 4, 7 } Point-set of 2-(7, 4, 2) Design with 7 blocks > Supp := Support(b); > Supp, Universe(Supp); { 2, 3, 4, 7 } Integer Ring