Elementary Invariants of an Incidence Structure

All operations defined for incidence structures apply also to near–linear spaces, linear spaces and designs.

NumberOfPoints(D): Inc -> RngInt

# P: IncPtSet -> RngIntElt

The cardinality \(v\) of the point set \(P\) of the incidence structure \(D\).

Points(D): Inc -> { IncPt }

An indexed set \(E\) whose elements are the points of the incidence structure \(D\). Note that this creates a standard set and not the point-set of \(D\), in contrast to the function PointSet.

Support(D): Inc -> { Elt }

An indexed set \(E\) which is the underlying point set of the incidence structure \(D\) (i.e., the elements of the set have their “real” types; they are no longer from the category IncPt).

PointDegrees(D): Inc -> [ RngIntElt ]

A sequence whose \(i\)-th term gives the number of blocks containing the \(i\)-th point of the design \(D\).

NumberOfBlocks(D): Inc -> RngIntElt

# B: IncBlkSet -> RngIntElt

The number of blocks \(b\) of the incidence structure \(D\) with block-set \(B\).

Blocks(D): Inc -> { IncBlk}

An indexed set containing the blocks of the incidence structure \(D\). In contrast to the function BlockSet, this function returns the collection of blocks of \(D\) in the form of a standard set.

BlockDegrees(D): Inc -> [ RngIntElt ]

BlockSizes(D): Inc -> [ RngIntElt ]

A sequence whose \(i\)-th term gives the number of points in the \(i\)-th block of the incidence structure \(D\).

Covalence(D, S): Inc, { IncPt } -> RngIntElt

Given a subset \(S\) of the point set of an incidence structure \(D\), return the number of blocks of \(D\) that contain \(S\).

IncidenceMatrix(D): Inc -> ModMatRngElt

The incidence matrix of the incidence structure \(D\).

pRank(D, p): Inc, RngIntElt -> RngIntElt

The \(p\)-rank of the incidence structure \(D\).