Structures and Congruences

Oliver Kullmann

Mathematical structures

Equivalence Structures and Congruences relations

Subspaces and quotients Oliver Kullmann theorems Computer Science Department Congruences for Swansea University groups and rings

MRes Seminar Swansea, November 10, 2008 Structures and Introduction Congruences Oliver Kullmann This lecture tries to convey some key concepts from Mathematical mathematics in general, relevant for our investigations structures Equivalence into boolean algebra and Stone duality: relations First we reflect upon “mathematical structures”. Subspaces and quotients

Of fundamental importance is the concept of an Isomorphisms “compatible” with the structure, theorems Congruences for which enables us to derive a “quotient”. groups and rings We discuss this concept in general, and especially for the algebraic structures here. Basically no proofs are given, but emphasise is put on the definitions and properties, leaving the proofs as relatively straightforward exercises.

You must fill the gaps yourself!! Structures and Overview Congruences Oliver Kullmann

Mathematical structures

1 Mathematical structures Equivalence relations

Subspaces and quotients 2 Equivalence relations Isomorphisms theorems

Congruences for 3 Subspaces and quotients groups and rings

4 Isomorphisms theorems

5 Congruences for groups and rings Structures and Three major mathematical structures Congruences Oliver Kullmann Bourbaki introduced “structuralism” to mathematics (and Mathematical the world), distinguishing three major types of structures mathematical structures: Equivalence relations

order structures (“less”, “maximal”, inequalities, ...) Subspaces and quotients algebraic structures (operations, equalities, ...) Isomorphisms topological structures (“near”, “continuity”, ...). theorems Congruences for These structures occur in pure and in mixed forms. In groups and rings each of the basic cases, a structure is a pair

(M, S)

where M is the “base ” (the set of objects of the structure under consideration), while S is some set whose elements are constructed from the elements of M (pairs, tuples or for example). Structures and Isomorphisms and morphisms Congruences Oliver Kullmann The structure S on the base set specifies the relations Mathematical between the elements of the base set. structures 0 0 Equivalence Between different structures (M, S), (M , S ) we relations always have the notion of an , a Subspaces and quotients bijection f : M → M0 such that “transporting” S by f Isomorphisms (“renaming” the elements, leaving the set-structures theorems unchanged) yields S0 (one writes (M, S) =∼ (M0, S0)). Congruences for groups and rings In “normal” mathematical situation we also have the more general notion of a morphism, which are maps f : M → M0 which “preserve” the structures S, S0. Typically this means that forward resp. backward transport of S resp. S0 yields a of S0 resp. S. Isomorphisms then are bijective maps which are morphisms in both directions. Structures and Structures of order Congruences Oliver Kullmann

The most important order structure is the Mathematical structures

Equivalence partially ordered set (“poset”), relations

pairs (M, ≤) where ≤ is a partial order on M. Subspaces and quotients (Recall the characteristic properties of partial orders: Isomorphisms theorems reflexivity, antisymmetry, transitivity.) Congruences for Morphisms from (M, ≤) to (M0, ≤0) are isoton maps, groups and rings i.e., f : M → M0 such that x ≤ y ⇒ f (x) ≤ f (y). A (harmless) generalisation is that of a quasi-ordered set (“qoset”), where antisymmetry is dropped. Isoton maps are just given by forward-transportation, i.e., for (x, y) ∈ ≤ we have (f (x), f (y)) ∈ ≤0. Structures and Algebraic structures Congruences Oliver Kullmann The most fundamental is that of a Mathematical groupoid (or “magma”): structures Equivalence A pair (M, ◦), where ◦ : M × M → M relations Subspaces and (i.e., ◦ is an internal binary composition). quotients

Isomorphisms Morphisms from (M, ◦) to (M0, ◦0) are theorems 0 Congruences for , i.e., f : M → M such that groups and rings f (x ◦ y) = f (x) ◦0 f (y). Important specialisations impose conditions like associativity, commutativity and/or neutral elements and invertibility. Homomorphisms are again just given by forward-transportation, i.e., for ((x, y), z) ∈ ◦ we have ((f (x), f (y)), f (z)) ∈ ◦0. Structures and Topological structures Congruences Oliver Kullmann

Mathematical The most important topological structure is that of a structures Equivalence topological space, which are relations

Subspaces and pairs (M, τ), where τ is a set of subsets of M quotients closed under arbitrary unions and finite intersections. Isomorphisms theorems

Congruences for Morphisms from (M, τ) to (M0, τ 0) are continuous groups and rings maps, i.e., f : M → M0 such that for O ∈ τ 0 we have f −1(O) ∈ τ. Isomorphisms are now called homeomorphisms. So, this time continuous maps are just given by backward-transportation. Structures and Remark: Forward and backward Congruences Oliver Kullmann ( , ) A very general type of structure is a pair M S , where S Mathematical is an arbitrary set of subsets of M. For morphisms we structures Equivalence have the two standard possibilities: relations 0 1 Backward morphisms f : M → M from (M, S) to Subspaces and quotients (M0, 0) are given by the condition S Isomorphisms 0 − S ∈ S ⇒ f 1(S) ∈ S. theorems Congruences for 2 0 Forward morphisms f : M → M from (M, S) to groups and rings 0 0 0 (M , S ) are given by the condition S ∈ S ⇒ f (S) ∈ S . Using backward morphisms, we have as important special cases the mathematical structures of topological spaces and “measurable spaces”. Using forward morphisms, we obtain the fundamental combinatorial structure of (infinite) hypergraphs (with infinite hyperedges). Structures and Evil combinatorics Congruences Oliver Kullmann Recall: For a map f , the inverse relation f −1 preserves Mathematical the basic set-theoretical operations (union, intersection, structures

set-difference, etc.), while f itself does not! Equivalence relations

So we get “beautiful” topology and measure theory, and Subspaces and “ugly” combinatorics. quotients Isomorphisms theorems

It would be interesting to investigate this fundamental Congruences for difference in a general way. (For example, for finite groups and rings 0 0 (M, S), (M , S ) the existence of a forward morphism is NP-complete — what about backward morphisms? Perhaps further conditions on the set systems are needed, to give the backward-morphisms an edge — every concrete mathematical structure together with their morphisms can be represented by (arbitrary) set systems with forward morphisms as well as by (arbitrary) set systems with backward morphisms.) Structures and Prototypes Congruences Oliver Kullmann

Mathematical Injectivity and surjectivity are properties of maps of structures Equivalence fundamental importance. relations The “prototype” of an injective map is the canonical Subspaces and quotients embedding Isomorphisms i : A ,→ X theorems Congruences for of a subset A ⊆ X into the base set X (given as the groups and rings restriction of the identity map of X). We are now seeking for the corresponding “prototype” π : X  B of surjective maps. Structures and Equivalence relations and partitions Congruences Oliver Kullmann

Mathematical Recall: An equivalence relation on a set X is a relation structures ∼ ⊆ X which is Equivalence relations

reflexive (on X; i.e., ∀ x ∈ X : x ∼ x) Subspaces and quotients symmetric (i.e., ∀ x, y : x ∼ y ⇒ y ∼ x) Isomorphisms transitive (i.e., ∀ x, y, z : x ∼ y ∧ y ∼ z ⇒ x ∼ z). theorems Congruences for So equivalence relations are special (extreme) groups and rings quasi-orders. A partition of X is a set P of subsets of X such that P covers X (i.e., S P = X) P is disjoint (∀ A, B ∈ P : A ∩ B = ∅) the elements of P are non-empty. Structures and Correspondence between equivalence Congruences relations and partitions Oliver Kullmann Mathematical The equivalence relations on X are in one-to-one structures

correspondence to the partitions P of X: Equivalence relations Given an equivalence relation ∼ on X, we define the Subspaces and [x]∼ of an element x ∈ X as the quotients set of elements equivalent to x: Isomorphisms theorems

0 0 Congruences for [x]∼ := {x ∈ X : x ∼ X } groups and rings

Now the set {[x]∼ : x ∈ X} of equivalence classes is a partition of X. Given a partition P of X, the corresponding equivalence relation ∼P makes two elements equivalent if they are in the same “part” of P:

x ∼P y :⇔ ∃ P ∈ P : {x, y} ⊆ P. Structures and Maps and equivalence relations Congruences Oliver Kullmann Every map f : X → Y induces an equivalence relation Mathematical ∼f on X, by making two elements equivalent if they get structures

the same value: Equivalence relations ∼ := {(x, x0) ∈ X 2 : f (x) = f (x0)}. Subspaces and f quotients

Isomorphisms On the other hand, every equivalence relation ∼ on X theorems induces the canonical surjection π∼, mapping every Congruences for groups and rings element to its equivalence class:

X/∼ := {[x]∼ : x ∈ X}

π∼ : X → X/∼

π∼(x) := [x]∼.

The canonical surjections are the prototypes of surjective maps. Structures and Inducing structures on subsets Congruences Oliver Kullmann

Consider a structure (M, S) and a subset A ⊆ M. Mathematical structures The task is to define an induced structure SA, such Equivalence relations that (A, SA) is of the same type (“species”). Subspaces and Minimal conditions are: quotients S = S Isomorphisms M theorems for B ⊆ A: (SA)B = SB. Congruences for The standard procedure is to restrict S to A, where the groups and rings following tweaks need to be considered: 1 If the species involves some totality condition, then only certain “closed” A are suitable. 2 If backward morphisms are used, then inverse images regarding the canonical embeddings are to be used. Structures and Morphism conditions for induced structures Congruences Oliver Kullmann

Mathematical structures Typically the following two conditions provide necessary Equivalence relations and sufficient conditions for the “right” induced structures: Subspaces and quotients

Isomorphisms 1 The canonical embedding iA : A → M is a morphism theorems from (A, SA) to (M, S). Congruences for groups and rings 2 Given any structure (M0, S0) (from the species), a map f : M0 → A is a morphism from (M0, S0) to 0 0 (A, SA) iff iA ◦ f is a morphism from (M , S ) to (M, S). The second condition expresses that whether we consider A on its own or as a part of M shouldn’t matter. Structures and Induced partial orders Congruences Oliver Kullmann

Mathematical structures For a partial order (M, ≤) and A ⊆ M the induced partial Equivalence relations

order ≤A is Subspaces and 2 quotients ≤A:= ≤ ∩ A . Isomorphisms theorems

Congruences for In other words, we restrict ≤ to A. groups and rings This induced structure fulfils all conditions we mentioned. And the induced structure is determined by the morphism conditions. Structures and Subgroupoids Congruences Oliver Kullmann

Given a groupoid (M, ◦) and A ⊆ M, the natural attempt is Mathematical structures

to use Equivalence ◦ ∩ (A2 × A) relations Subspaces and (using that ◦ ⊆ M2 × M). quotients Isomorphisms However in this way we only obtain a partial theorems 0 Congruences for groupoid, since for a, a ∈ A in general we do not groups and rings have a ◦ a0 ∈ A. So we only consider closed A. Then again all conditions are fulfilled. Here already the first morphism condition determines the induced structure (while the second condition comes for free). Structures and Induced topologies Congruences Oliver Kullmann

Given a topological space (M, τ) and A ⊆ M, a first Mathematical structures attempt would be to define τA as the set of all O ∈ τ with Equivalence O ⊆ A, but in general this is not even a topology. relations

Subspaces and Since backward morphisms are used for topology, the quotients

correct definition is to use all subsets of A which could be Isomorphisms induced by O ∈ τ, that is, theorems Congruences for groups and rings τA := {O ∩ A : O ∈ τ}.

Now again all conditions are fulfilled. And the induced topology is determined by the morphism conditions. The induced topology is the coarsest topology such that the canonical embedding is continuous. Structures and Inducing structures on quotients Congruences Consider structure (M, S) and equivalence ∼ on M. Oliver Kullmann

The task is to define a quotient structure S/∼ , s.t. Mathematical structures (A, S/∼ ) is a structure of the same species. Equivalence Minimal conditions are: relations Subspaces and S/ idx = S quotients ∼0 /∼ for an equivalence relation on S : Isomorphisms (S/∼ )/∼ 0 =∼ S/(∼0 ◦ ∼) (via the canonical bijection). theorems 0 Congruences for 0 Here ∼ ◦ ∼ has the canonical surjection π∼ ◦ π∼. groups and rings The standard procedure is to transfer S to M/∼ , where the following tweaks need to be considered: 1 If the species involves some uniqueness condition, then only certain “compatible” ∼ are suitable. 2 For backward morphisms inverse images regarding the canonical surjections are to be used. 3 The quotient formation might violate certain separation conditions for the species. Structures and Morphism conditions for quotient structures Congruences Oliver Kullmann

Mathematical Typically the following two conditions provide necessary structures Equivalence and sufficient conditions for the “right” quotient structures: relations

Subspaces and quotients 1 The canonical surjection π∼ : M → M/∼ is a Isomorphisms morphism from (M, S) to (M/∼ , S∼). theorems Congruences for 2 0 0 Given any structure (M , S ) (from the species), a groups and rings 0 map f : M/∼ → M is a morphism from (M/∼ , S∼) 0 0 to (M , S ) iff f ◦ π∼ is a morphism from (M, S) to (M0, S0). The second condition expresses that the abstraction provided by the quotient formation is transparent if we are handling maps which are compatible with that abstraction. Structures and Quotients of partial orders Congruences Oliver Kullmann

Mathematical structures ( , ≤) ∼ Equivalence For a partial order M and an equivalence relation relations

on M the quotient order ≤ /∼ is Subspaces and quotients 0 0 0 0 ≤ /∼ := {([x]∼, [y]∼) | ∃ x ∼ x ∃ y ∼ y : x ≤ y }. Isomorphisms theorems

Congruences for groups and rings The quotient order in general is only a quasi-order, but otherwise fulfils all conditions we mentioned. And the quotient structure is determined by the morphisms conditions. Structures and Quotients of groupoids Congruences Oliver Kullmann

Given a groupoid (M, ◦) and an equivalence relation ∼ on Mathematical structures M, the natural attempt is to use the mapping Equivalence ([x]∼, [y]∼) 7→ [x ◦ y]∼, however in general this yields a relations Subspaces and multivalued operation. quotients So we need to consider equivalence relations ∼ Isomorphisms theorems which are compatible with ◦. Such equivalence Congruences for relations are called congruence relations. groups and rings ∼ is a congruence relation for (M, ◦) iff for x ∼ x0 and y ∼ y 0 we have x ◦ y ∼ x0 ◦ y 0. Then again all conditions are fulfilled. Here (again) already the first morphism condition determines the quotient structure (while the second condition comes for free). Structures and Quotient topologies Congruences Oliver Kullmann

Mathematical structures Given a topological space (M, τ) and an equivalence Equivalence relations relation ∼ on M, the quotient topology is Subspaces and [ quotients τ/∼ := {O ⊆ M/∼ : O ∈ τ}. Isomorphisms theorems

Congruences for Now again all conditions are fulfilled. groups and rings And the quotient topology is determined by the morphism conditions. The quotient topology is the finest topology such that the canonical surjection is continuous. Structures and Embeddings and quotient maps Congruences Oliver Kullmann Consider a morphism f :(M, S) → (M0, S0) between two Mathematical structures: structures Equivalence f is called an embedding if f is an isomorphism of relations 0 0 (M, S) to the of f (as sub-structure of (M , S )). Subspaces and quotients The induced equivalence relation ∼:=∼ is always f Isomorphisms compatible with the structure on M (thus yields a theorems congruence relations for algebraic structures). Congruences for groups and rings f is called a quotient map if f yields (in the natural way) an isomorphism from (M, S)/∼ to (M0, S0) (note that f is compatible with∼ , and thus always yields a morphism from (M, S)/∼ to (M0, S0)). The natural examples are: Canonical embeddings are embeddings. Canonical surjections are quotient maps. Structures and Characterisations of embeddings and Congruences quotient maps Oliver Kullmann Mathematical structures

Equivalence relations For algebraic structures we have: Subspaces and quotients

f is an embedding iff f is injective; Isomorphisms theorems

f is a quotient map iff f is surjective. Congruences for However order structures and topological structures have groups and rings more possibilities (they are less rigid w.r.t. formation of substructures and quotient structures), and thus here injectivity resp. surjectivity of f is only necessary but not sufficient (in general). Structures and Strengthening “transitivity” of induced Congruences structures Oliver Kullmann Mathematical structures

Equivalence We can strengthen the already stated “transitivity relations Subspaces and conditions” to characterise all substructures resp. quotient quotients

structures of substructures resp. quotient structures: Isomorphisms theorems

Given a structure (M, S) and a substructure (A, SA), Congruences for groups and rings the substructures of (A, SA) are exactly the substructures (B, SB) of (M, S) with B ⊆ A. And for a compatible equivalence relation ∼, the compatible equivalence relations of (M, S)/∼ are exactly the compatible equivalence relations ∼0 of (M, S) which are courser than ∼. Structures and Kernels Congruences Oliver Kullmann

Mathematical A congruence relation ∼ in an arbitrary groupoid (V , ◦) is structures Equivalence a wild beast. We need more structure to tame it. relations A unital groupoid is a triple (V , ◦, e) such that e is a Subspaces and quotients (and thus the) neutral element of (V , ◦). Morphisms Isomorphisms of unital groupoids by definition respect the neutral theorems element. Congruences for groups and rings Given a f :(V , ◦, e) → (V 0, ◦0, e0), the equivalence class of e w.r.t. ∼f is the of f (the inverse image of e0). The kernel ker(f ) is a sub-unital-groupoid. In general the kernel does not determine (by far) the congruence relation induced by f . Structures and Kernels in groups — normal subgroups Congruences Oliver Kullmann

Mathematical More structure for groupoids: structures

1 Equivalence A is an associative groupoid. relations 2 A is a unital groupoid (V , ◦, e) such that Subspaces and quotients (V , ◦) is a semigroup. Isomorphisms 3 A is a monoid where every element has an theorems Congruences for inverse. groups and rings The kernel of a f : G → G0 has the special structure of being a , i.e., for all group elements g ∈ G we have g ker(f )g−1 = ker(f ).

And from the kernel ker(f ) we can reconstruct ∼f by −1 x ∼f y ⇔ x ◦ y ∈ ker(f ). Structures and Congruence relations in groups by normal Congruences subgroups Oliver Kullmann Mathematical So we have seen that every congruence relation ∼ in structures

a group determines a (unique) normal subgroup (as Equivalence relations the equivalence class of the neutral element). Subspaces and Conversely, given a normal subgroup N of a group quotients G, we obtain a congruence relation ∼ by Isomorphisms theorems −1 x ∼ y :⇔ x ◦ y ∈ N. Congruences for So for groups we have the (powerful) situation that groups and rings congruence relations are in one-to-one correspondence to special subgroups, namely normal subgroups. Given a normal subgroup N, for the quotient G/N the whole (normal) subgroup N becomes the (new) neutral element, and this identification of N to a single point can consistently be extended to the whole group: x = y mod N ⇐⇒ xy −1 = e mod N ⇐⇒ xy −1 ∈ N. Structures and Kernels in rings — ideals Congruences Oliver Kullmann For commutative groups every subgroup is normal, so Mathematical that here the congruence relations just correspond to the structures

subgroups. Equivalence relations A is a quintuple (R, +, ·, 0, 1) such that (R, +, 0) Subspaces and is a commutative group, (R, ·, 1) is a monoid, and we quotients have both distributivity laws. Isomorphisms theorems

Congruence relations now have to take care also of Congruences for the (in general not commutative). groups and rings It is easy to see that the role of normal subgroups for groups is now played by ideals: An in R is an additive subgroup I such that for all r ∈ R we have r · I ⊆ I. The ideals in rings are exactly the kernels of (arbitrary) ring homomorphisms, and the congruence relations in rings correspond one-to-one to the ideals of the ring. Structures and Congruences

Oliver Kullmann

Mathematical structures

Equivalence relations

Subspaces and quotients

Isomorphisms End theorems Congruences for (for now) groups and rings