Resolution (block plan)

A resolution of a 2- block plan (a special incidence structure ) is a generalization of the parallelism of block plans in finite geometry . The partition of the set of d -dimensional subspaces as blocks of an affine geometry in sets of parallels is a 1-resolution of this geometry as a 2-block plan. A block plan that allows a resolution is called a dissolvable block plan , with this resolution the block set breaks down into a maximum number c of generalized sets of parallels , then one speaks of a strong dissolution and calls the block plan strongly dissolvable . ${\displaystyle AG_{d}(n,q)}$

Definitions

• Be a block plan. A resolution of is a partition of the block set of into flocks such that there are positive integers with the property that each point in lies within exactly blocks of . The numbers are called the parameters of the resolution. If all parameters of a resolution are the same , one speaks of a resolution. ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \{\rho _{1},\ldots ,\rho _{c}\}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle \rho _{i}}$ ${\displaystyle {\mathfrak {B}}_{i}}$ ${\displaystyle \rho _{1},\ldots ,\rho _{c}}$ ${\displaystyle \rho }$ ${\displaystyle \rho }$
• A block plan is called dissolvable or - dissolvable if it has a resolution or a resolution. ${\displaystyle \rho }$ ${\displaystyle \rho }$
• If a resolvable block plan with c classes is valid , then this resolution is called strong resolution of the block plan and the block plan is called strongly resolvable . ${\displaystyle {\mathcal {I}}}$ ${\displaystyle b+1=v+c}$
• If two blocks of a resolvable block diagram are in the same class , then one also writes and names the blocks in parallel with regard to the resolution . The generalized parallelism defined in this way is obviously an equivalence relation on the set of blocks. ${\displaystyle B,C\in {\mathfrak {B}}_{i}}$ ${\displaystyle {\mathfrak {B}}_{i}}$ ${\displaystyle B\parallel C}$
• For a resolution you set the number of blocks in the family . ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle m_{i}=|{\mathfrak {B}}_{i}|,1\leq i\leq c}$ ${\displaystyle {\mathfrak {B}}_{i}}$

properties

Be a block diagram that has a resolution with the parameters . Then applies ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \rho _{1},\ldots ,\rho _{c}}$

1. ${\displaystyle m_{i}\cdot k=v\cdot \rho _{i},\quad (1\leq i\leq c),}$
2. ${\displaystyle \rho _{1}+\cdots +\rho _{c}=r,\quad m_{1}+\cdots +m_{c}=b,}$
3. If has a -resolution, then k is a divisor of and each class has the same number m of blocks. ${\displaystyle {\mathcal {I}}}$ ${\displaystyle \rho }$ ${\displaystyle v\cdot \rho }$

• If there is a resolvable block diagram with c classes, then is . A strong resolution is therefore a resolution with the block for the amount of maximum number of crowds. ${\displaystyle {\mathcal {I}}}$ ${\displaystyle b+1\geq v+c}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathcal {I}}}$

Hughes and Piper theorem about strong resolutions

• The following theorem by Hughes and Piper characterizes the strong resolutions:

Let be a block diagram with b blocks that has a resolution . Then and equality applies if and only if there are two non-negative numbers ("inner number of cuts") and ("outer number of cuts") with the following properties: ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle b+1\geq v+c}$ ${\displaystyle \mu _{I}}$ ${\displaystyle \mu _{A}}$

• Every two different blocks of the same class always have exactly intersections and ${\displaystyle \mu _{I}}$
• Every two blocks from different classes always have exactly intersections. ${\displaystyle \mu _{A}}$

Beker's Theorem about Solvable 3-Block Plans

• The set of Beker clarifies the question of when a highly resolvable block plan is a three-block plan:

The highly resolvable 3-block plans are exactly the Hadamard 3-block plans .

Examples

• Each block diagram has the trivial resolution , i.e. H. every block plan can be r resolved. - In a block diagram, the number indicates with how many blocks a given point is incised. ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle \{{\mathfrak {B}}\}}$ ${\displaystyle r=b_{1}}$
• If there is a dissolution of , then a dissolution of is obtained again when certain groups are united to form a new group. For example, and again are resolutions of . ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle \{{\mathfrak {B}}_{1}\cup {\mathfrak {B}}_{2},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \{{\mathfrak {B}}_{1}\cup \ldots \cup {\mathfrak {B}}_{c-1},{\mathfrak {B}}_{c}\}}$ ${\displaystyle {\mathcal {I}}}$
• A block diagram can only be 1-resolved if it has parallelism. The resolution is the division of the block quantity into sets of parallel and it applies that the inner number of cuts is then , but the outer number of cuts need not be constant. ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle c=r=b_{1}}$ ${\displaystyle \mu _{I}=0}$

• In particular, an affine geometry with its usual parallelism is 1-solvable and it then applies , that is, the number of parallels in each family is the same, the outer number of intersections is constant, if the block set is the set of hyperplanes of space. ${\displaystyle AG_{d}(n,q)}$ ${\displaystyle m_{i}=b/r}$ ${\displaystyle d=n-1}$
• Every affine block plan can be 1-resolved due to its parallelism, here too is the same for every set of parallels. ${\displaystyle m_{i}=b/r}$

Generalization: Tactical Decomposition

Each resolution of a 2-block plan also provides a special tactical breakdown of this block plan. With this generalization of the concept of “dissolving a block plan”, in addition to the partitioning of the block set into (generalized parallels) families, the point set is also broken down into several “point classes” .

literature

Articles on individual questions

• Daniel R. Hughes, Fred C. Piper: On resolutions and Bose's theorem . In: Geom. Dedicata . tape 5 , 1976, p. 129-133 , doi : 10.1007 / BF00148147 .
• Henry Beker: On strong tactical decompositions . In: Journal of the London Mathematical Society . tape 16 , 1977, pp. 191–196 ( abstract [accessed May 2, 2013]).

Textbooks

• Albrecht Beutelspacher : Introduction to finite geometry I . Block plans. Bibliographische Institut, Mannheim / Vienna / Zurich / New York 1982, ISBN 3-411-01632-9 , Chapter 5. Resolutions and decompositions, pp. 196–240.
• Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . BI Wissenschaftsverlag, Mannheim 1986, ISBN 0-521-33334-2 .
• DR Hughes, FC Piper: Projective planes . Springer, Berlin / Heidelberg / New York 1973 (Here the solvability is only defined and examined for the special cases of affine geometries.).

Individual evidence

1. ↑ ^{a b c d} Beutelspacher (1982)
2. ↑ Beutelspacher (1982), Lemma 5.1.1
3. ↑ Beutelspacher (1982), Corollary 5.1.2
4. ↑ Hughes, Piper (1976); Beutelspacher (1982), main clause 5.1.9
5. ↑ Beker (1977)
6. ↑ Beutelspacher (1982), sentence 5.1.10