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Structures and Congruences Structures and Congruences Oliver Kullmann Mathematical structures Equivalence Structures and Congruences relations Subspaces and quotients Oliver Kullmann Isomorphisms 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 equivalence relation “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 set” (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 subsets 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 isomorphism, 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 subset 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 algebraic structure 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 homomorphisms, 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 equivalence class [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”).
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