# 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 $D$.