Sigma-algebraFrom Wikipedia, the free encyclopedia

Chapter 1

Delta-ring

In mathematics, a nonempty collection of setsR is called a δ-ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A ∪B ∈ R if A,B ∈ R

2. A−B ∈ R if A,B ∈ R

3.∩∞

n=1 An ∈ R if An ∈ R for all n ∈ N

If only the first two properties are satisfied, then R is a ring but not a δ-ring. Every σ-ring is a δ-ring, but not everyδ-ring is a σ-ring.δ-rings can be used instead of σ-fields in the development of measure theory if one does not wish to allow sets ofinfinite measure.

1.1 See also• Ring of sets

• Sigma field

• Sigma ring

1.2 References• Cortzen, Allan. “Delta-Ring.” From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.http://mathworld.wolfram.com/Delta-Ring.html

2

Chapter 2

Field of sets

“Set algebra” redirects here. For the basic properties and laws of sets, see Algebra of sets.

In mathematics a field of sets is a pair ⟨X,F⟩ whereX is a set andF is an algebra over X i.e., a non-empty subsetof the power set of X closed under the intersection and union of pairs of sets and under complements of individualsets. In other words F forms a subalgebra of the power set Boolean algebra of X . (Many authors refer to F itselfas a field of sets. The word “field” in “field of sets” is not used with the meaning of field from field theory.) Elementsof X are called points and those of F are called complexes and are said to be the admissible sets of X .Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can berepresented as a field of sets.

2.1 Fields of sets in the representation theory of Boolean algebras

2.1.1 Stone representation

Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each elementof the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power setrepresentation can be constructed more generally for any complete atomic Boolean algebra.In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representationby considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Booleanalgebra correspond to its ultrafilters and that an atom is below an element of a finite Boolean algebra if and only ifthat element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of aBoolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Booleanalgebra the set of ultrafilters containing that element. This construction does indeed produce a representation of theBoolean algebra as a field of sets and is known as the Stone representation. It is the basis of Stone’s representationtheorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or filters,similar to Dedekind cuts.Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexesby associating each element of the Boolean algebra with the set of such homomorphisms that map it to the topelement. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the topelements under these homomorphisms.) With this approach one sees that Stone representation can also be regardedas a generalization of the representation of finite Boolean algebras by truth tables.

2.1.2 Separative and compact fields of sets: towards Stone duality

• A field of sets is called separative (or differentiated) if and only if for every pair of distinct points there is acomplex containing one and not the other.

• Afield of sets is called compact if and only if for every proper filter overX the intersection of all the complexes

3

4 CHAPTER 2. FIELD OF SETS

contained in the filter is non-empty.

These definitions arise from considering the topology generated by the complexes of a field of sets. Given a field ofsets X = ⟨X,F⟩ the complexes form a base for a topology, we denote the corresponding topological space by T (X). Then

• T (X) is always a zero-dimensional space.

• T (X) is a Hausdorff space if and only if X is separative.

• T (X) is a compact space with compact open sets F if and only if X is compact.

• T (X) is a Boolean space with clopen sets F if and only if X is both separative and compact (in which case itis described as being descriptive)

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space isknown as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexesof the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stonerepresentation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a dualityexists between Boolean algebras and Boolean spaces.

2.2 Fields of sets with additional structure

2.2.1 Sigma algebras and measure spaces

If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebraand the corresponding field of sets is called a measurable space. The complexes of a measurable space are calledmeasurable sets.A measure space is a triple ⟨X,F , µ⟩ where ⟨X,F⟩ is a measurable space and µ is a measure defined on it. If µis in fact a probability measure we speak of a probability space and call its underlying measurable space a samplespace. The points of a sample space are called samples and represent potential outcomes while the measurable sets(complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Manyuse the term sample space simply for the underlying set of a probability space, particularly in the case where everysubset is an event.) Measure spaces and probability spaces play a foundational role in measure theory and probabilitytheory respectively.The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

2.2.2 Topological fields of sets

A topological field of sets is a triple ⟨X, T ,F⟩ where ⟨X, T ⟩ is a topological space and ⟨X,F⟩ is a field of setswhich is closed under the closure operator of T or equivalently under the interior operator i.e. the closure and interiorof every complex is also a complex. In other words F forms a subalgebra of the power set interior algebra on ⟨X, T ⟩.Every interior algebra can be represented as a topological field of sets with its interior and closure operators corre-sponding to those of the topological space.Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interiorand closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets.

Algebraic fields of sets and Stone fields

A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes.If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets areprecisely the open complexes. Moreover the open complexes form a base for the topology.

2.2. FIELDS OF SETS WITH ADDITIONAL STRUCTURE 5

Topological fields of sets that are separative, compact and algebraic are calledStone fields and provide a generalizationof the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation ofits underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated bythe complexes corresponding to the open elements of the interior algebra (which form a base for a topology). Thesecomplexes are then precisely the open complexes and the construction produces a Stone field representing the interioralgebra - the Stone representation.

2.2.3 Preorder fields

A preorder field is a triple ⟨X,≤,F⟩ where ⟨X,≤⟩ is a preordered set and ⟨X,F⟩ is a field of sets.Like the topological fields of sets, preorder fields play an important role in the representation theory of interior alge-bras. Every interior algebra can be represented as a preorder field with its interior and closure operators correspondingto those of the Alexandrov topology induced by the preorder. In other words

Int(S) = x ∈ X : there exists a y ∈ S with y ≤ x andCl(S) = x ∈ X : there exists a y ∈ S with x ≤ y for all S ∈ F

Preorder fields arise naturally in modal logic where the points represent the possible worlds in the Kripke semanticsof a theory in the modal logic S4 (a formal mathematical abstraction of epistemic logic), the preorder represents theaccessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worldsin which individual sentences in the theory hold, providing a representation of the Lindenbaum-Tarski algebra of thetheory.

Algebraic and canonical preorder fields

A preorder field is called algebraic if and only if it has a set of complexes A which determines the preorder in thefollowing manner: x ≤ y if and only if for every complex S ∈ A , x ∈ S implies y ∈ S . The preorder fieldsobtained from S4 theories are always algebraic, the complexes determining the preorder being the sets of possibleworlds in which the sentences of the theory closed under necessity hold.A separative compact algebraic preorder field is said to be canonical. Given an interior algebra, by replacing thetopology of its Stone representation with the corresponding canonical preorder (specialization preorder) we obtain arepresentation of the interior algebra as a canonical preorder field. By replacing the preorder by its correspondingAlexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. (Thetopology of this "Alexandrov representation" is just the Alexandrov bi-coreflection of the topology of the Stonerepresentation.)

2.2.4 Complex algebras and fields of sets on relational structures

The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbi-trary (normal) Boolean algebras with operators. For this we consider structures ⟨X, (Ri)I ,F⟩ where ⟨X, (Ri)I⟩ is arelational structure i.e. a set with an indexed family of relations defined on it, and ⟨X,F⟩ is a field of sets. The com-plex algebra (or algebra of complexes) determined by a field of sets X = ⟨X, (Ri)I ,F⟩ on a relational structure,is the Boolean algebra with operators

C(X) = ⟨F ,∩,∪, ′, ∅, X, (fi)I⟩

where for all i ∈ I , if Ri is a relation of arity n+ 1 , then fi is an operator of arity n and for all S1, ..., Sn ∈ F

fi(S1, ..., Sn) = x ∈ X : there exist x1 ∈ S1, ..., xn ∈ Sn such that Ri(x1, ..., xn, x)

This construction can be generalized to fields of sets on arbitrary algebraic structures having both operators andrelations as operators can be viewed as a special case of relations. If F is the whole power set of X then C(X) iscalled a full complex algebra or power algebra.

6 CHAPTER 2. FIELD OF SETS

Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in thesense that it is isomorphic to the complex algebra corresponding to the field.(Historically the term complex was first used in the case where the algebraic structure was a group and has its originsin 19th century group theory where a subset of a group was called a complex.)

2.3 See also• List of Boolean algebra topics

• Algebra of sets

• Sigma algebra

• Measure theory

• Probability theory

• Interior algebra

• Alexandrov topology

• Stone’s representation theorem for Boolean algebras

• Stone duality

• Boolean ring

• Preordered field

2.4 References• Goldblatt, R., Algebraic Polymodal Logic: A Survey, Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450,July 2000

• Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, 44, p. 173-242, 1989

• Johnstone, Peter T. (1982). Stone spaces (3rd ed.). Cambridge: Cambridge University Press. ISBN 0-521-33779-8.

• Naturman, C.A., Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Math-ematics, 1991

• Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., Handbook of Modal Logic, Volume 3 ofStudies in Logic and Practical Reasoning, Elsevier, 2006

2.5 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the finite intersectionproperty (FIP) if the intersection over any finite subcollection of A is nonempty. It has the strong finite intersectionproperty (SFIP) if the intersection over any finite subcollection of A is infinite.A centered system of sets is a collection of sets with the finite intersection property.

3.1 Definition

LetX be a set withA = Aii∈I a family of subsets ofX . Then the collectionA has the finite intersection property(FIP), if any finite subcollection J ⊆ I has non-empty intersection

∩i∈J Ai.

3.2 Discussion

Clearly the empty set cannot belong to any collection with the finite intersection property. The condition is triviallysatisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satisfied if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any finite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits.The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact ifand only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself.[1]This formulation of compactness is used in some proofs of Tychonoff’s theorem and the uncountability of the realnumbers (see next section)

3.3 Applications

Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U ⊆ X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV ⊂ U whose closure doesn’t contain x (x may or may not be in U). Choose y in U different from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isn’t in U, this is possible sinceU is nonempty). Then by the Hausdorff condition, choose disjoint neighbourhoodsW and K of x and y respectively.Then K ∩ U will be a neighbourhood of y contained in U whose closure doesn’t contain x as desired.

Now suppose f : N → X is a bijection, and let xi : i ∈ N denote the image of f. Let X be the first open set

7

8 CHAPTER 3. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 ⊂ X whose closure doesn’t contain x1. Secondly, choose a neighbourhood U2 ⊂U1 whose closure doesn’t contain x2. Continue this process whereby choosing a neighbourhood Un₊₁ ⊂ Un whoseclosure doesn’t contain xn₊₁. Then the collection Ui : i ∈ N satisfies the finite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesn’t belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdorff condition; a countable set with the indiscrete topology is compact, has morethan one point, and satisfies the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a finite space as the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdorff space is uncountable.Proof. Let X be a perfect, compact, Hausdorff space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdorff space which is not compact, then the one-point compactification of X isa perfect, compact Hausdorff space. Therefore the one point compactification of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

3.4 Examples

A filter has the finite intersection property by definition.

3.5 Theorems

Let X be nonempty, F ⊆ 2X, F having the finite intersection property. Then there exists an F′ ultrafilter (in 2X) suchthat F ⊆ F′.See details and proof in Csirmaz & Hajnal (1994).[2] This result is known as ultrafilter lemma.

3.6 Variants

A family of sets A has the strong finite intersection property (sfip), if every finite subfamily of A has infiniteintersection.

3.7 References[1] A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.org.

[2] Csirmaz, László; Hajnal, András (1994), Matematikai logika (In Hungarian), Budapest: Eötvös Loránd University.

• Finite intersection property at PlanetMath.org.

Chapter 4

Helly family

In combinatorics, a Helly family of order k is a family of sets such that any minimal subfamily with an emptyintersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k -fold intersection is non-empty has non-empty total intersection.[1]

The k-Helly property is the property of being a Helly family of order k.[2] These concepts are named after EduardHelly (1884 - 1943); Helly’s theorem on convex sets, which gave rise to this notion, states that convex sets in Euclideanspace of dimension n are a Helly family of order n + 1.[1] The number k is frequently omitted from these names inthe case that k = 2.

4.1 Examples

• In the family of all subsets of the set a,b,c,d, the subfamily a,b,c, a,b,d, a,c,d, b,c,d has an emptyintersection, but removing any set from this subfamily causes it to have a nonempty intersection. Therefore,it is a minimal subfamily with an empty intersection. It has four sets in it, and is the largest possible minimalsubfamily with an empty intersection, so the family of all subsets of the set a,b,c,d is a Helly family of order4.

• Let I be a finite set of closed intervals of the real line with an empty intersection. Let A be the interval whose leftendpoint a is as large as possible, and let B be the interval whose right endpoint b is as small as possible. Then,if a were less than or equal to b, all numbers in the range [a,b] would belong to all invervals of I, violatingthe assumption that the intersection of I is empty, so it must be the case that a > b. Thus, the two-intervalsubfamily A,B has an empty intersection, and the family I cannot be minimal unless I = A,B. Therefore,all minimal families of intervals with empty intersections have two or fewer intervals in them, showing that theset of all intervals is a Helly family of order 2.[3]

• The family of infinite arithmetic progressions of integers also has the 2-Helly property. That is, whenever afinite collection of progressions has the property that no two of them are disjoint, then there exists an integerthat belongs to all of them; this is the Chinese remainder theorem.[2]

4.2 Formal definition

More formally, a Helly family of order k is a set system (F, E), with F a collection of subsets of E, such that, forevery finite G ⊆ F with

∩X∈G

X = ∅,

we can find H ⊆ G such that

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10 CHAPTER 4. HELLY FAMILY

∩X∈H

X = ∅

and

|H| ≤ k. [1]

In some cases, the same definition holds for every subcollection G, regardless of finiteness. However, this is a morerestrictive condition. For instance, the open intervals of the real line satisfy the Helly property for finite subcollections,but not for infinite subcollections: the intervals (0,1/i) (for i = 0, 1, 2, ...) have pairwise nonempty intersections, buthave an empty overall intersection.

4.3 Helly dimension

If a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension ofa metric space is one less than the Helly number of the family of metric balls in that space; Helly’s theorem impliesthat the Helly dimension of a Euclidean space equals its dimension as a real vector space.[4]

The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number ofthe family of translates of S.[5] For instance, the Helly dimension of any hypercube is 1, even though such a shapemay belong to a Euclidean space of much higher dimension.[6]

Helly dimension has also been applied to other mathematical objects. For instance Domokos (2007) defines the Hellydimension of a group (an algebraic structure formed by an invertible and associative binary operation) to be one lessthan the Helly number of the family of left cosets of the group.[7]

4.4 The Helly property

If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k forwhich the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as theHelly property. A 2-Hellyfamily is also known as a Helly family.[1][2]

A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, inthe stronger variant of Helly dimension for infinite subcollections) is called injective or hyperconvex.[8] The existenceof the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.[9]

4.5 References[1] Bollobás, Béla (1986), Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, Cam-

bridge University Press, p. 82, ISBN 9780521337038.

[2] Duchet, Pierre (1995), “Hypergraphs”, in Graham, R. L.; Grötschel, M.; Lovász, L., Handbook of combinatorics, Vol. 1,2, Amsterdam: Elsevier, pp. 381–432, MR 1373663. See in particular Section 2.5, “Helly Property”, pp. 393–394.

[3] This is the one-dimensional case of Helly’s theorem. For essentially this proof, with a colorful phrasing involving sleep-ing students, see Savchev, Svetoslav; Andreescu, Titu (2003), “27 Helly’s Theorem for One Dimension”, MathematicalMiniatures, New Mathematical Library 43, Mathematical Association of America, pp. 104–106, ISBN 9780883856451.

[4] Martini, Horst (1997), Excursions Into Combinatorial Geometry, Springer, pp. 92–93, ISBN 9783540613411.

[5] Bezdek, Károly (2010), Classical Topics in Discrete Geometry, Springer, p. 27, ISBN 9781441906007.

[6] Sz.-Nagy, Béla (1954), “Ein Satz über Parallelverschiebungen konvexer Körper”, Acta Universitatis Szegediensis 15: 169–177, MR 0065942.

[7] Domokos,M. (2007), “Typical separating invariants”, TransformationGroups 12 (1): 49–63, arXiv:math/0511300, doi:10.1007/s00031-005-1131-4, MR 2308028.

4.5. REFERENCES 11

[8] Deza, Michel Marie; Deza, Elena (2012), Encyclopedia of Distances, Springer, p. 19, ISBN 9783642309588

[9] Isbell, J. R. (1964), “Six theorems about injectivemetric spaces”, Comment. Math. Helv. 39: 65–76, doi:10.1007/BF02566944.

Chapter 5

Ring of sets

Not to be confused with Ring (mathematics).

In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In ordertheory, a nonempty family of setsR is called a ring (of sets) if it is closed under intersection and union. That is, thefollowing two statements are true for all sets A and B ,

1. A,B ∈ R implies A ∩B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R. [1]

In measure theory, a ring of setsR is instead a nonempty family closed under unions and set-theoretic differences.[2]That is, the following two statements are true for all sets A and B (including when they are the same set),

1. A,B ∈ R implies A \B ∈ R and

2. A,B ∈ R implies A ∪B ∈ R.

This implies the empty set is in R . It also implies that R is closed under symmetric difference and intersection,because of the identities

1. AB = (A \B) ∪ (B \A) and

2. A ∩B = A \ (A \B).

(So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, theseoperations give R the structure of a boolean ring. Conversely, every family of sets closed under both symmetricdifference and intersection is also closed under union and differences. This is due to the identities

1. A ∪B = (AB) (A ∩B) and

2. A \B = A (A ∩B).

5.1 Examples

If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.If (X,≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongsto an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions. However,in general it will not be closed under differences of sets.The open sets and closed sets of any topological space are closed under both unions and intersections.[1]

12

5.2. RELATED STRUCTURES 13

On the real line R, the family of sets consisting of the empty set and all finite unions of intervals of the form (a, b],a,b in R is a ring in the measure theory sense.If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed underboth unions and intersections.[1]

If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form aring of sets.[1]

5.2 Related structures

A ring of sets (in the order-theoretic sense) forms a distributive lattice in which the intersection and union operationscorrespond to the lattice’s meet and join operations, respectively. Conversely, every distributive lattice is isomorphicto a ring of sets; in the case of finite distributive lattices, this is Birkhoff’s representation theorem and the sets maybe taken as the lower sets of a partially ordered set.[1]

A field of subsets of X is a ring that contains X and is closed under relative complement. Every field, and so alsoevery σ-algebra, is a ring of sets in the measure theory sense.A semi-ring (of sets) is a family of sets S with the properties

1. ∅ ∈ S,

2. A,B ∈ S implies A ∩B ∈ S, and

3. A,B ∈ S implies A \B =∪n

i=1 Ci for some disjoint C1, . . . , Cn ∈ S.

Clearly, every ring (in the measure theory sense) is a semi-ring.A semi-field of subsets of X is a semi-ring that contains X.

5.3 References[1] Birkhoff, Garrett (1937), “Rings of sets”,DukeMathematical Journal 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-

X, MR 1546000.

[2] De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.

5.4 External links• Ring of sets at Encyclopedia of Mathematics

Chapter 6

Sigma-algebra

"Σ-algebra” redirects here. For an algebraic structure admitting a given signature Σ of operations, see Universal al-gebra.

In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a setX is a collection Σ of subsets of X that is closed under countable-fold set operations (complement, union of countablymany sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitelymany set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations.The pair (X, Σ) is also a field of sets, called a measurable space.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for whicha given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as thefoundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events whichcan be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.In statistics, (sub) σ-algebras are needed for a formal mathematical definition of sufficient statistic,[1] particularlywhen the statistic is a function or a random process and the notion of conditional density is not applicable.If X = a, b, c, d, one possible σ-algebra on X is Σ = ∅, a, b, c, d, a, b, c, d , where ∅ is the empty set.However, a finite algebra is always a σ-algebra.If A1, A2, A3, … is a countable partition of X then the collection of all unions of sets in the partition (includingthe empty set) is a σ-algebra.A more useful example is the set of subsets of the real line formed by starting with all open intervals and addingin all countable unions, countable intersections, and relative complements and continuing this process (by transfiniteiteration through all countable ordinals) until the relevant closure properties are achieved (a construction known asthe Borel hierarchy).

6.1 Motivation

There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managingpartial information characterized by sets.

6.1.1 Measure

Ameasure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as makingprecise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of theirindividual sizes, even for an infinite sequence of disjoint sets.One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example theaxiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the realline, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead asmaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

14

6.1. MOTIVATION 15

operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable setand the countable union of measurable sets is a measurable set. Non-empty collections of sets with these propertiesare called σ-algebras.

6.1.2 Limits of sets

Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences ofsets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows onσ-algebras.

• The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim supn→∞

An =∞∩

n=1

∞∪m=n

Am.

• The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim infn→∞

An =∞∪

n=1

∞∩m=n

Am.

• If, in fact,

lim infn→∞

An = lim supn→∞

An

then the limn→∞ An exists as that common set.

6.1.3 Sub σ-algebras

In much of probability, especially when conditional expectation is involved, one is concerned with sets that representonly part of all the possible information that can be observed. This partial information can be characterized with asmaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only toand determined only by the partial information. A simple example suffices to illustrate this idea.Imagine you are playing a game that involves flipping a coin repeatedly and observing whether it comes up Heads (H)or Tails (T). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last.This means the sample space Ω must consist of all possible infinite sequences of H or T :

Ω = H,T∞ = (x1, x2, x3, . . . ) : xi ∈ H,T, i ≥ 1

However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the nextflip. The observed information at that point can be described in terms of the 2n possibilities for the first n flips.Formally, since you need to use subsets of Ω, this is codified as the σ-algebra

Gn = A× H,T∞ : A ⊂ H,Tn

Observe that then

G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G∞

where G∞ is the smallest σ-algebra containing all the others.

16 CHAPTER 6. SIGMA-ALGEBRA

6.2 Definition and properties

6.2.1 Definition

Let X be some set, and let 2X represent its power set. Then a subset Σ ⊂ 2X is called a σ-algebra if it satisfies thefollowing three properties:[2]

1. X is in Σ, and X is considered to be the universal set in the following context.

2. Σ is closed under complementation: If A is in Σ, then so is its complement, X\A.

3. Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1 ∪ A2 ∪ A3 ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying DeMorgan’s laws).It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set,is also in Σ. Moreover, by (3) it follows as well that X, ∅ is the smallest possible σ-algebra.Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebraover X, is called a measurable space. A function between two measurable spaces is called a measurable function ifthe preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with themeasurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞].A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin’s theorem(below).

6.2.2 Dynkin’s π-λ theorem

This theorem (or the related monotone class theorem) is an essential tool for proving many results about propertiesof specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

A π-system P is a collection of subsets of Σ that is closed under finitely many intersections, anda Dynkin system (or λ-system) D is a collection of subsets of Σ that contains Σ and is closed undercomplement and under countable unions of disjoint subsets.

Dynkin’s π-λ theorem says, if P is a π-system and D is a Dynkin system that contains P then the σ-algebra σ(P)generated by P is contained in D. Since certain π-systems are relatively simple classes, it may not be hard to verifythat all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of allsubsets with the property is a Dynkin system can also be straightforward. Dynkin’s π-λ Theorem then implies thatall sets in σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P).One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures orintegrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integraltypically associated with computing the probability:

P(X ∈ A) =∫AF (dx) for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while P is a probability measure, defined ona σ-algebra Σ of subsets of some sample space Ω.

6.2.3 Combining σ-algebras

Suppose Σα : α ∈ A is a collection of σ-algebras on a space X.

• The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it oftenis denoted by:

6.2. DEFINITION AND PROPERTIES 17

∧α∈A

Σα.

Sketch of Proof: Let Σ∗ denote the intersection. Since X is in every Σα, Σ∗ is not empty. Closure undercomplement and countable unions for every Σα implies the same must be true for Σ∗. Therefore, Σ∗ is aσ-algebra.

• The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates aσ-algebra known as the join which typically is denoted

∨α∈A

Σα = σ

( ∪α∈A

Σα

).

A π-system that generates the join is

P =

n∩

i=1

Ai : Ai ∈ Σαi , αi ∈ A, n ≥ 1

.

Sketch of Proof: By the case n = 1, it is seen that each Σα ⊂ P , so∪α∈A

Σα ⊂ P.

This implies

σ

( ∪α∈A

Σα

)⊂ σ(P)

by the definition of a σ-algebra generated by a collection of subsets. On the other hand,

P ⊂ σ

( ∪α∈A

Σα

)

which, by Dynkin’s π-λ theorem, implies

σ(P) ⊂ σ

( ∪α∈A

Σα

).

6.2.4 σ-algebras for subspaces

Suppose Y is a subset of X and let (X, Σ) be a measurable space.

• The collection Y ∩ B: B ∈ Σ is a σ-algebra of subsets of Y.

• Suppose (Y, Λ) is a measurable space. The collection A ⊂ X : A ∩ Y ∈ Λ is a σ-algebra of subsets of X.

6.2.5 Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[3] A σ-ring need not be a σ-algebra, as for examplemeasurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real linehas infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takesmeasurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtainedby their countable union yet its measure is not finite.

18 CHAPTER 6. SIGMA-ALGEBRA

6.2.6 Typographic note

σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, Σ) may bedenoted as (X,F) or (X,F) .

6.3 Examples

6.3.1 Simple set-based examples

Let X be any set.

• The family consisting only of the empty set and the set X, called the minimal or trivial σ-algebra over X.

• The power set of X, called the discrete σ-algebra.

• The collection ∅, A, Ac, X is a simple σ-algebra generated by the subset A.

• The collection of subsets of X which are countable or whose complements are countable is a σ-algebra (whichis distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by thesingletons of X. Note: “countable” includes finite or empty.

• The collection of all unions of sets in a countable partition of X is a σ-algebra.

6.3.2 Stopping time sigma-algebras

A stopping time τ can define a σ -algebraFτ , the so-called stopping time sigma-algebra, which in a filtered probabilityspace describes the information up to the random time τ in the sense that, if the filtered probability space is interpretedas a random experiment, the maximum information that can be found out about the experiment from arbitrarily oftenrepeating it until the time τ is Fτ .[4]

6.4 σ-algebras generated by families of sets

6.4.1 σ-algebra generated by an arbitrary family

Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every setin F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated byF.For a simple example, consider the set X = 1, 2, 3. Then the σ-algebra generated by the single subset 1 isσ(1) = ∅, 1, 2, 3, 1, 2, 3. By an abuse of notation, when a collection of subsets contains only oneelement, A, one may write σ(A) instead of σ(A); in the prior example σ(1) instead of σ(1). Indeed, usingσ(A1, A2, ...) to mean σ(A1, A2, ...) is also quite common.There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

6.4.2 σ-algebra generated by a function

If f is a function from a set X to a set Y and B is a σ-algebra of subsets of Y, then the σ-algebra generated by thefunction f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B. i.e.

σ(f) = f−1(S) |S ∈ B.

A function f from a set X to a set Y is measurable with respect to a σ-algebra Σ of subsets of X if and only if σ(f) isa subset of Σ.

6.4. Σ-ALGEBRAS GENERATED BY FAMILIES OF SETS 19

One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topologicalspace and B is the collection of Borel sets on Y.If f is a function from X to Rn then σ(f) is generated by the family of subsets which are inverse images of inter-vals/rectangles in Rn:

σ(f) = σ(f−1((a1, b1]× · · · × (an, bn]) : ai, bi ∈ R

).

A useful property is the following. Assume f is a measurable map from (X, ΣX) to (S, ΣS) and g is a measurable mapfrom (X, ΣX) to (T, ΣT). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂σ(g). If S is finite or countably infinite or if (S, ΣS) is a standard Borel space (e.g., a separable complete metric spacewith its associated Borel sets) then the converse is also true.[5] Examples of standard Borel spaces include Rn with itsBorel sets and R∞ with the cylinder σ-algebra described below.

6.4.3 Borel and Lebesgue σ-algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or,equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivialexample that is not a Borel set, see the Vitali set or Non-Borel sets.On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebracontains more sets than the Borel σ-algebra onRn and is preferred in integration theory, as it gives a complete measurespace.

6.4.4 Product σ-algebra

Let (X1,Σ1) and (X2,Σ2) be two measurable spaces. The σ-algebra for the corresponding product spaceX1 ×X2

is called the product σ-algebra and is defined by

Σ1 × Σ2 = σ(B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2).

Observe that B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2 is a π-system.The Borel σ-algebra for Rn is generated by half-infinite rectangles and by finite rectangles. For example,

B(Rn) = σ ((−∞, b1]× · · · × (−∞, bn] : bi ∈ R) = σ ((a1, b1]× · · · × (an, bn] : ai, bi ∈ R) .

For each of these two examples, the generating family is a π-system.

6.4.5 σ-algebra generated by cylinder sets

Suppose

X ⊂ RT = f : f(t) ∈ R, t ∈ T

is a set of real-valued functions. Let B(R) denote the Borel subsets ofR. A cylinder subset of X is a finitely restrictedset defined as

Ct1,...,tn(B1, . . . , Bn) = f ∈ X : f(ti) ∈ Bi, 1 ≤ i ≤ n.

Each

Ct1,...,tn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n

20 CHAPTER 6. SIGMA-ALGEBRA

is a π-system that generates a σ-algebra Σt1,...,tn . Then the family of subsets

FX =

∞∪n=1

∪ti∈T,i≤n

Σt1,...,tn

is an algebra that generates the cylinder σ-algebra for X. This σ-algebra is a subalgebra of the Borel σ-algebradetermined by the product topology of RT restricted to X.An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In thiscase, it suffices to consider the cylinder sets

Cn(B1, . . . , Bn) = (B1 × · · · ×Bn × R∞) ∩X = (x1, x2, . . . , xn, xn+1, . . . ) ∈ X : xi ∈ Bi, 1 ≤ i ≤ n,

for which

Σn = σ(Cn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n)

is a non-decreasing sequence of σ-algebras.

6.4.6 σ-algebra generated by random variable or vector

Suppose (Ω,Σ,P) is a probability space. If Y : Ω → Rn is measurable with respect to the Borel σ-algebra on Rn

then Y is called a random variable (n = 1) or random vector (n ≥ 1). The σ-algebra generated by Y is

σ(Y ) = Y −1(A) : A ∈ B(Rn).

6.4.7 σ-algebra generated by a stochastic process

Suppose (Ω,Σ,P) is a probability space and RT is the set of real-valued functions on T . If Y : Ω → X ⊂ RT ismeasurable with respect to the cylinder σ-algebra σ(FX) (see above) for X then Y is called a stochastic process orrandom process. The σ-algebra generated by Y is

σ(Y ) =Y −1(A) : A ∈ σ(FX)

= σ(Y −1(A) : A ∈ FX),

the σ-algebra generated by the inverse images of cylinder sets.

6.5 See also

• Join (sigma algebra)

• Measurable function

• Sample space

• Separable sigma algebra

• Sigma ring

• Sigma additivity

6.6. REFERENCES 21

6.6 References[1] Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1-118-12237-2.

[2] Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

[3] Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 978-0-470-31795-2.

[4] Fischer, Tom (2013). “On simple representations of stopping times and stopping time sigma-algebras”. Statistics andProbability Letters 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.

[5] Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

6.7 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Sigma Algebra from PlanetMath.

Chapter 7

Sigma-ideal

In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read “sigma,” means countable in thiscontext) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent applicationis perhaps in probability theory.Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if thefollowing properties are satisfied:(i) Ø ∈ N;(ii) When A ∈ N and B ∈ Σ , B ⊆ A⇒ B ∈ N;(iii) Ann∈N ⊆ N ⇒

∪n∈N An ∈ N.

Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. Theconcept of σ-ideal is dual to that of a countably complete (σ-) filter.If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.The notion can be generalized to preorders (P,≤,0) with a bottom element 0 as follows: I is a σ-ideal of P just when(i') 0 ∈ I,(ii') x ≤ y & y ∈ I ⇒ x ∈ I, and(iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ≤ y for each nThus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist).It is natural in this context to ask that P itself have countable suprema.

7.1 References• Bauer, Heinz (2001): Measure and Integration Theory. Walter de Gruyter GmbH & Co. KG, 10785 Berlin,Germany.

22

Chapter 8

Sigma-ring

Inmathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countableunion and relative complementation.

8.1 Formal definition

LetR be a nonempty collection of sets. ThenR is a σ-ring if:

1.∪∞

n=1 An ∈ R if An ∈ R for all n ∈ N

2. A \B ∈ R if A,B ∈ R

8.2 Properties

From these two properties we immediately see that

∩∞n=1 An ∈ R if An ∈ R for all n ∈ N

This is simply because ∩∞n=1An = A1 \ ∪∞

n=1(A1 \An) .

8.3 Similar concepts

If the first property is weakened to closure under finite union (i.e.,A∪B ∈ RwheneverA,B ∈ R ) but not countableunion, thenR is a ring but not a σ-ring.

8.4 Uses

σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one doesnot wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be aσ-field.A σ-ring R that is a collection of subsets of X induces a σ-field for X . Define A to be the collection of all subsetsof X that are elements ofR or whose complements are elements ofR . Then A is a σ-field over the set X . In factA is the minimal σ-field containingR since it must be contained in every σ-field containingR .

23

24 CHAPTER 8. SIGMA-RING

8.5 See also• Delta ring

• Ring of sets

• Sigma field

8.6 References• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-ringsin development of Lebesgue theory.

Chapter 9

Steiner system

The Fano plane is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongsto a unique line.

In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically at-design with λ = 1 and t ≥ 2.

25

26 CHAPTER 9. STEINER SYSTEM

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsetsof S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In an alternatenotation for block designs, an S(t,k,n) would be a t-(n,k,1) design.This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition requiredthat k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple (or triad) system, while an S(3,4,n) was called aSteiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictlyadhered to.A long-standing problem in design theory is if any nontrivial (t < k < n) Steiner systems have t ≥ 6; also if infinitelymany have t = 4 or 5.[1] This was claimed to be solved in the affirmative by Peter Keevash.[2][3]

9.1 Examples

9.1.1 Finite projective planes

A finite projective plane of order q, with the lines as blocks, is an S(2, q + 1, q2 + q + 1) , since it has q2 + q + 1points, each line passes through q + 1 points, and each pair of distinct points lies on exactly one line.

9.1.2 Finite affine planes

A finite affine plane of order q, with the lines as blocks, is an S(2, q, q2). An affine plane of order q can be obtainedfrom a projective plane of the same order by removing one block and all of the points in that block from the projectiveplane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.

9.2 Classical Steiner systems

9.2.1 Steiner triple systems

An S(2,3,n) is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviationSTS(n) for a Steiner triple system of order n. The number of triples is n(n−1)/6. A necessary and sufficient conditionfor the existence of an S(2,3,n) is that n ≡ 1 or 3 (mod 6). The projective plane of order 2 (the Fano plane) is anSTS(7) and the affine plane of order 3 is an STS(9).Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829STS(19)s.[4]

We can define a multiplication on the set S using the Steiner triple system by setting aa = a for all a in S, and ab= c if a,b,c is a triple. This makes S an idempotent, commutative quasigroup. It has the additional property that“ab” = “c” implies “bc” = “a” and “ca” = “b”.[5] Conversely, any (finite) quasigroup with these properties arises froma Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called Steinerquasigroups.[6]

9.2.2 Steiner quadruple systems

An S(3,4,n) is called a Steiner quadruple system. A necessary and sufficient condition for the existence of anS(3,4,n) is that n ≡ 2 or 4 (mod 6). The abbreviation SQS(n) is often used for these systems.Up to isomorphism, SQS(8) and SQS(10) are unique, there are 4 SQS(14)s and 1,054,163 SQS(16)s.[7]

9.2.3 Steiner quintuple systems

An S(4,5,n) is called a Steiner quintuple system. A necessary condition for the existence of such a system is that n≡ 3or 5 (mod 6) which comes from considerations that apply to all the classical Steiner systems. An additional necessarycondition is that n ≡ 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficientconditions are not known.

9.3. PROPERTIES 27

There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17.[8] Systems are known fororders 23, 35, 47, 71, 83, 107, 131, 167 and 243. The smallest order for which the existence is not known (as of2011) is 21.

9.3 Properties

It is clear from the definition of S(t,k,n) that 1 < t < k < n . (Equalities, while technically possible, lead to trivialsystems.)If S(t,k,n) exists, then taking all blocks containing a specific element and discarding that element gives a derived systemS(t−1,k−1,n−1). Therefore the existence of S(t−1,k−1,n−1) is a necessary condition for the existence of S(t,k,n).The number of t-element subsets in S is

(nt

), while the number of t-element subsets in each block is

(kt

). Since

every t-element subset is contained in exactly one block, we have(nt

)= b(kt

), or b =

(nt

)(kt

) , where b is the numberof blocks. Similar reasoning about t-element subsets containing a particular element gives us

(n−1t−1

)= r

(k−1t−1

),

or r =

(n−1t−1

)(k−1t−1

) , where r is the number of blocks containing any given element. From these definitions follows the

equation bk = rn . It is a necessary condition for the existence of S(t,k,n) that b and r are integers. As with anyblock design, Fisher’s inequality b ≥ n is true in Steiner systems.Given the parameters of a Steiner system S(t,k,n) and a subset of size t′ ≤ t , contained in at least one block, onecan compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascaltriangle.[9] In particular, the number of blocks intersecting a fixed block in any number of elements is independent ofthe chosen block.It can be shown that if there is a Steiner system S(2,k,n), where k is a prime power greater than 1, then n ≡ 1 or k(mod k(k−1)). In particular, a Steiner triple system S(2,3,n) must have n = 6m+1 or 6m+3. It is known that this is theonly restriction on Steiner triple systems, that is, for each natural number m, systems S(2,3,6m+1) and S(2,3,6m+3)exist.

9.4 History

Steiner triple systems were defined for the first time by W.S.B. Woolhouse in 1844 in the Prize question #1733 ofLady’s and Gentlemen’s Diary.[10] The posed problem was solved by Thomas Kirkman (1847). In 1850 Kirkmanposed a variation of the problem known as Kirkman’s schoolgirl problem, which asks for triple systems having anadditional property (resolvability). Unaware of Kirkman’s work, Jakob Steiner (1853) reintroduced triple systems,and as this work was more widely known, the systems were named in his honor.

9.5 Mathieu groups

Several examples of Steiner systems are closely related to group theory. In particular, the finite simple groups calledMathieu groups arise as automorphism groups of Steiner systems:

• The Mathieu group M11 is the automorphism group of a S(4,5,11) Steiner system

• The Mathieu group M12 is the automorphism group of a S(5,6,12) Steiner system

• The Mathieu group M22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steinersystem

• The Mathieu group M23 is the automorphism group of a S(4,7,23) Steiner system

• The Mathieu group M24 is the automorphism group of a S(5,8,24) Steiner system.

28 CHAPTER 9. STEINER SYSTEM

9.6 The Steiner system S(5, 6, 12)

There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M12, and in that context itis denoted by W12.

9.6.1 Constructions

There are different ways to construct an S(5,6,12) system.

Projective line method

This construction is due to Carmichael (1937).[11]

Add a new element, call it ∞, to the 11 elements of the finite field F11 (that is, the integers mod 11). This set, S,of 12 elements can be formally identified with the points of the projective line over F11. Call the following specificsubset of size 6,

∞, 1, 3, 4, 5, 9,

a “block”. From this block, we obtain the other blocks of the S(5,6,12) system by repeatedly applying the linearfractional transformations:

z′ = f(z) =az + b

cz + dwhere ,a, b, c, d in are F11 and ad− bc in square non-zero a is F11.

With the usual conventions of defining f (−d/c) = ∞ and f (∞) = a/c, these functions map the set S onto itself. Ingeometric language, they are projectivities of the projective line. They form a group under composition which is theprojective special linear group PSL(2,11) of order 660. There are exactly five elements of this group that leave thestarting block fixed setwise,[12] so there will be 132 images of that block. As a consequence of the multiply transitiveproperty of this group acting on this set, any subset of five elements of S will appear in exactly one of these 132images of size six.

Kitten method

An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis,[13] which was intended as a “handcalculator” to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid ofnumbers, which represent an affine geometry on the vector space F3xF3, an S(2,3,9) system.

Construction from K6 graph factorization

The relations between the graph factors of the complete graph K6 generate an S(5,6,12).[14] AK6 graph has 6 different1-factorizations (ways to partition the edges into disjoint perfect matchings), and also 6 vertices. The set of verticesand the set of factorizations provide one block each. For every distinct pair of factorizations, there exists exactly oneperfect matching in common. Take the set of vertices and replace the two vertices corresponding to an edge of thecommon perfect matching with the labels corresponding to the factorizations; add that to the set of blocks. Repeatthis with the other two edges of the common perfect matching. Similarly take the set of factorizations and replacethe labels corresponding to the two factorizations with the end points of an edge in the common perfect matching.Repeat with the other two edges in the matching. There are thus 3+3 = 6 blocks per pair of factorizations, and thereare 6C2 = 15 pairs among the 6 factorizations, resulting in 90 new blocks. Finally take the full set of 12C6 = 924combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any ofthe 92 blocks generated so far. Exactly 40 blocks remain, resulting in 2+90+40 = 132 blocks of the S(5,6,12).

9.7. THE STEINER SYSTEM S(5, 8, 24) 29

9.7 The Steiner system S(5, 8, 24)

The Steiner system S(5, 8, 24), also known as theWitt design orWitt geometry, was first described by Carmichael(1931) and rediscovered by Witt (1938). This system is connected with many of the sporadic simple groups and withthe exceptional 24-dimensional lattice known as the Leech lattice.The automorphism group of S(5, 8, 24) is the Mathieu group M24, and in that context the design is denoted W24

(“W” for “Witt”)

9.7.1 Constructions

There are many ways to construct the S(5,8,24). Two methods are described here:

Method based on 8-combinations of 24 elements

All 8-element subsets of a 24-element set are generated in lexicographic order, and any such subset which differsfrom some subset already found in fewer than four positions is discarded.The list of octads for the elements 01, 02, 03, ..., 22, 23, 24 is then:

01 02 03 04 05 06 07 0801 02 03 04 09 10 11 1201 02 03 04 13 14 15 16.. (next 753 octads omitted).13 14 15 16 17 18 19 2013 14 15 16 21 22 23 2417 18 19 20 21 22 23 24

Each single element occurs 253 times somewhere in some octad. Each pair occurs 77 times. Each triple occurs 21times. Each quadruple (tetrad) occurs 5 times. Each quintuple (pentad) occurs once. Not every hexad, heptad oroctad occurs.

Method based on 24-bit binary strings

All 24-bit binary strings are generated in lexicographic order, and any such string that differs from some earlier onein fewer than 8 positions is discarded. The result looks like this:000000000000000000000000 000000000000000011111111 000000000000111100001111 000000000000111111110000000000000011001100110011 000000000011001111001100 000000000011110000111100 000000000011110011000011000000000101010101010101 000000000101010110101010 . . (next 4083 24-bit strings omitted) . 111111111111000011110000111111111111111100000000 111111111111111111111111The list contains 4096 items, which are each code words of the extended binary Golay code. They form a groupunder the XOR operation. One of them has zero 1-bits, 759 of them have eight 1-bits, 2576 of them have twelve1-bits, 759 of them have sixteen 1-bits, and one has twenty-four 1-bits. The 759 8-element blocks of the S(5,8,24)(called octads) are given by the patterns of 1’s in the code words with eight 1-bits.

9.8 See also• Constant weight code

• Kirkman’s schoolgirl problem

• Sylvester–Gallai configuration

30 CHAPTER 9. STEINER SYSTEM

9.9 Notes[1] “Encyclopaedia of Design Theory: t-Designs”. Designtheory.org. 2004-10-04. Retrieved 2012-08-17.

[2] Keevash, Peter (2014). “The existence of designs”. arXiv:1401.3665.

[3] “A Design Dilemma Solved, Minus Designs”. Quanta Magazine. 2015-06-09. Retrieved 2015-06-27.

[4] Colbourn & Dinitz 2007, pg.60

[5] This property is equivalent to saying that (xy)y = x for all x and y in the idempotent commutative quasigroup.

[6] Colbourn & Dinitz 2007, pg. 497, definition 28.12

[7] Colbourn & Dinitz 2007, pg.106

[8] Östergard & Pottonen 2008

[9] Assmus & Key 1994, pg. 8

[10] Lindner & Rodger 1997, pg.3

[11] Carmichael 1956, p. 431

[12] Beth, Jungnickel & Lenz 1986, p. 196

[13] Curtis 1984

[14] EAGTS textbook

9.10 References• Assmus, E. F., Jr.; Key, J. D. (1994), “8. Steiner Systems”, Designs and Their Codes, Cambridge UniversityPress, pp. 295–316, ISBN 0-521-45839-0.

• Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

• Carmichael, Robert (1931), “Tactical Configurations of Rank Two”, American Journal of Mathematics 53:217–240, doi:10.2307/2370885

• Carmichael, Robert D. (1956) [1937], Introduction to the theory of Groups of Finite Order, Dover, ISBN 0-486-60300-8

• Colbourn, Charles J.; Dinitz, Jeffrey H. (1996), Handbook of Combinatorial Designs, Boca Raton: Chapman& Hall/ CRC, ISBN 0-8493-8948-8, Zbl 0836.00010

• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8, Zbl 1101.05001

• Curtis, R.T. (1984), “The Steiner system S(5,6,12), the Mathieu group M12 and the “kitten"", in Atkinson,Michael D., Computational group theory (Durham, 1982), London: Academic Press, pp. 353–358, ISBN0-12-066270-1, MR 0760669

• Hughes, D. R.; Piper, F. C. (1985), Design Theory, Cambridge University Press, pp. 173–176, ISBN 0-521-35872-8.

• Kirkman, Thomas P. (1847), “On a Problem in Combinations”, The Cambridge and Dublin MathematicalJournal (Macmillan, Barclay, and Macmillan) II: 191–204.

• Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3• Östergard, Patric R.J.; Pottonen, Olli (2008), “There exists no Steiner system S(4,5,17)", Journal of Combina-torial Theory Series A 115 (8): 1570–1573, doi:10.1016/j.jcta.2008.04.005

• Steiner, J. (1853), “Combinatorische Aufgabe”, Journal für die Reine und Angewandte Mathematik 45: 181–182.

• Witt, Ernst (1938), “Die 5-Fach transitiven Gruppen von Mathieu”, Abh. Math. Sem. Univ. Hamburg 12:256–264, doi:10.1007/BF02948947

9.11. EXTERNAL LINKS 31

9.11 External links• Rowland, Todd and Weisstein, Eric W., “Steiner System”, MathWorld.

• Rumov, B.T. (2001), “Steiner system”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Steiner systems by Andries E. Brouwer

• Implementation of S(5,8,24) by Dr. Alberto Delgado, Gabe Hart, and Michael Kolkebeck

• S(5, 8, 24) Software and Listing by Johan E. Mebius

• The Witt Design computed by Ashay Dharwadker

32 CHAPTER 9. STEINER SYSTEM

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