Semirings Modeling Confidence and Uncertainty in
... A semiring S is called idempotent if, in addition to (S.1)–(S.4), its first operation is idempotent: (S.5) ∀ x ∈ S : x ⊕ x = x. A semiring S is called commutative if, in addition to (S.1)–(S.4), its second operation is commutative: (S.6) ∀ x, y ∈ S : x ⊗ y = y ⊗ x. Clearly, in a commutative semiring ...
... A semiring S is called idempotent if, in addition to (S.1)–(S.4), its first operation is idempotent: (S.5) ∀ x ∈ S : x ⊕ x = x. A semiring S is called commutative if, in addition to (S.1)–(S.4), its second operation is commutative: (S.6) ∀ x, y ∈ S : x ⊗ y = y ⊗ x. Clearly, in a commutative semiring ...
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1
... let us pick equivalence classes f ∈ Hom(x, y), g ∈ Hom(y, z) and representatives α : [0, 1] −→ X, β : [0, 1] −→ X of f and g, respectively. Then we define g ◦ f to be the equivalence class of β · α. Verify the following statements: (a) Composition of homotopy classes of continuous paths is associati ...
... let us pick equivalence classes f ∈ Hom(x, y), g ∈ Hom(y, z) and representatives α : [0, 1] −→ X, β : [0, 1] −→ X of f and g, respectively. Then we define g ◦ f to be the equivalence class of β · α. Verify the following statements: (a) Composition of homotopy classes of continuous paths is associati ...
Non-Homogenizable Classes of Finite Structures
... of regularity, which we call HN-regularity (we omit its technical definition), and prove in Theorem 3.1 from [14] that if F is a HN-regular class of finite connected structures, then Forbh (F) is homogenizable. In particular, if F is finite, then Forbh (F) is homogenizable. ~ and two I Example 2.1. ...
... of regularity, which we call HN-regularity (we omit its technical definition), and prove in Theorem 3.1 from [14] that if F is a HN-regular class of finite connected structures, then Forbh (F) is homogenizable. In particular, if F is finite, then Forbh (F) is homogenizable. ~ and two I Example 2.1. ...
Natural associativity and commutativity
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
... while associativity is an isomorphism a natural in its arguments A,B, and C. The general associative law again shows that any two iterated products F and F' of the n arguments A,, ...,A, are naturally isomorphic, under a natural isomorphism F z F' given by "iteration" of a. We then ask: what conditi ...
J. Harding, Orthomodularity of decompositions in a categorical
... 2. If f ⊕ g is defined and e ⊕ (f ⊕ g) is defined, then e ⊕ f is defined, (e ⊕ f ) ⊕ g is defined and e ⊕ (f ⊕ g) = (e ⊕ f ) ⊕ g. 3. For each f in X, there is exactly one f ∗ in X with f ⊕ f ∗ defined and f ⊕ f ∗ = 1. 4. If f ⊕ f is defined, then f = 0. Orthoalgebras were introduced by Randall and F ...
... 2. If f ⊕ g is defined and e ⊕ (f ⊕ g) is defined, then e ⊕ f is defined, (e ⊕ f ) ⊕ g is defined and e ⊕ (f ⊕ g) = (e ⊕ f ) ⊕ g. 3. For each f in X, there is exactly one f ∗ in X with f ⊕ f ∗ defined and f ⊕ f ∗ = 1. 4. If f ⊕ f is defined, then f = 0. Orthoalgebras were introduced by Randall and F ...
Composing functors Horizontal composition (functors): C D E If F, G
... Let Σ be a signature, and let E be a set of equations (both in the sense of universal algebra). A signature consists of a set |Σ| = {f, g, . . .} of function symbols, together with an assignment ar : |Σ| → N of an arity to each function symbol. Fix a signature. For example, let h be a function symbo ...
... Let Σ be a signature, and let E be a set of equations (both in the sense of universal algebra). A signature consists of a set |Σ| = {f, g, . . .} of function symbols, together with an assignment ar : |Σ| → N of an arity to each function symbol. Fix a signature. For example, let h be a function symbo ...
EXAMPLE SHEET 1 1. If k is a commutative ring, prove that b k
... is a C-comodule, write ρpxq “ ni“1 xi b ci with the ci linearly independent, and proceed in a similar way to the proof of the fundamental theorem of coalgebras). 16. Suppose that k is a field, and consider the functor Setop f Ñ Vect given on objects X by X ÞÑ k (Setf is the category of finite sets). ...
... is a C-comodule, write ρpxq “ ni“1 xi b ci with the ci linearly independent, and proceed in a similar way to the proof of the fundamental theorem of coalgebras). 16. Suppose that k is a field, and consider the functor Setop f Ñ Vect given on objects X by X ÞÑ k (Setf is the category of finite sets). ...
Equivalence relations handout
... equality in Z, but then moved on to talking about congruences, which is a weaker notion. This handout explains how “congruence modulo n” is something called an equivalence relation, and we can use it to construct a set Z/nZ that’s a “quotient” of Z. The key point is that a congruence modulo n in Z b ...
... equality in Z, but then moved on to talking about congruences, which is a weaker notion. This handout explains how “congruence modulo n” is something called an equivalence relation, and we can use it to construct a set Z/nZ that’s a “quotient” of Z. The key point is that a congruence modulo n in Z b ...
Math 8211 Homework 1 PJW
... isomorphic to the identity functor 1C and F G is naturally isomorphic to 1D (meaning that there are invertible natural transformations τ : GF → 1C and σ : F G → 1D ). 5. Let G and H be groups, which we may regard as categories with one object in which every morphism is invertible. (1) Show that a fu ...
... isomorphic to the identity functor 1C and F G is naturally isomorphic to 1D (meaning that there are invertible natural transformations τ : GF → 1C and σ : F G → 1D ). 5. Let G and H be groups, which we may regard as categories with one object in which every morphism is invertible. (1) Show that a fu ...
Categories and functors
... Definition 22.5. If C, D are preadditive categories, a functor F : C → D is called additive if F induces a homomorphism F : HomC (A, B) → HomD (F A, F B) for all A, B ∈ Ob(C). Lemma 22.6. An additive functor F : CR → Ab is the same as a left Rmodule. Proof. Suppose that M is a left R-module. Then M ...
... Definition 22.5. If C, D are preadditive categories, a functor F : C → D is called additive if F induces a homomorphism F : HomC (A, B) → HomD (F A, F B) for all A, B ∈ Ob(C). Lemma 22.6. An additive functor F : CR → Ab is the same as a left Rmodule. Proof. Suppose that M is a left R-module. Then M ...
Full text
... It follows from (1.9) and A(0,Q\a,$) = 1 that A(r,s\a,(3) is a polynomial in a,j3 and that the numerical coefficients in this polynomial are positive integers. Algebraic properties of A(r,s\a,fi) corresponding to the known properties of A(r,s) have been obtained in [ 3 ] ; also this paper includes a ...
... It follows from (1.9) and A(0,Q\a,$) = 1 that A(r,s\a,(3) is a polynomial in a,j3 and that the numerical coefficients in this polynomial are positive integers. Algebraic properties of A(r,s\a,fi) corresponding to the known properties of A(r,s) have been obtained in [ 3 ] ; also this paper includes a ...
9 Direct products, direct sums, and free abelian groups
... 12.4 Definition. Let {ci }i∈I be a family of objects in a category C. A (categorical) coproduct of the family {ci }i∈I is an object d ∈ C equipped with morphisms εi : ci → d for all i ∈ I that satisfies the following universal property. For any object b ∈ C and a family of morphisms {fi : ci → b}i∈I ...
... 12.4 Definition. Let {ci }i∈I be a family of objects in a category C. A (categorical) coproduct of the family {ci }i∈I is an object d ∈ C equipped with morphisms εi : ci → d for all i ∈ I that satisfies the following universal property. For any object b ∈ C and a family of morphisms {fi : ci → b}i∈I ...
Section I.3. Isomorphic Binary Structures
... is fundamentally different—in it, the binary operation when applied to a pair of the same elements yields that element (it is an idempotent binary operation—see page 28, Exercise 2.37). This is not the case in the first two tables and so ∗0 is not isomorphic to + nor ∗. ...
... is fundamentally different—in it, the binary operation when applied to a pair of the same elements yields that element (it is an idempotent binary operation—see page 28, Exercise 2.37). This is not the case in the first two tables and so ∗0 is not isomorphic to + nor ∗. ...
Amalgamation constructions in permutation group theory and model
... AP: take C to be the disjoint union of B1 and B2 over A with edges just those in B1 or B2 . (The free amalgam.) The Fraïssé limit of this amalgamation class is the random graph: it is the graph on vertex set N which you get with probability 1 by choosing independently with fixed probability p (6= 0, ...
... AP: take C to be the disjoint union of B1 and B2 over A with edges just those in B1 or B2 . (The free amalgam.) The Fraïssé limit of this amalgamation class is the random graph: it is the graph on vertex set N which you get with probability 1 by choosing independently with fixed probability p (6= 0, ...
Section I.7. Generating Sets and Cayley Digraphs
... Section I.7. Generating Sets and Cayley Digraphs Note. In this section, we generalize the idea of a single generator of a group to a whole set of generators of a group. Remember, a cyclic group has a single generator and is isomorphic to either Z (if it is of infinite order) or Zn (if it is of finit ...
... Section I.7. Generating Sets and Cayley Digraphs Note. In this section, we generalize the idea of a single generator of a group to a whole set of generators of a group. Remember, a cyclic group has a single generator and is isomorphic to either Z (if it is of infinite order) or Zn (if it is of finit ...
6 Permutation Groups - Arkansas Tech Faculty Web Sites
... A group of permutations , with composition as the operation, is called a permutation group on S. Example 6.1 1. Sym(S) is a permutation group. 2. The collection L of all invertible linear functions from R to R is a permutation group with respect to composition.(See Example 4.4.) Note that L is ...
... A group of permutations , with composition as the operation, is called a permutation group on S. Example 6.1 1. Sym(S) is a permutation group. 2. The collection L of all invertible linear functions from R to R is a permutation group with respect to composition.(See Example 4.4.) Note that L is ...
FINITE CATEGORIES WITH TWO OBJECTS A paper should have
... Lemma 4. If End(A) is finite and f ∈ End(A) satisfies left cancellation then f has a right inverse (a morphism h : A → A so that f ◦ h = idA . If both f and g satisfies left cancellation then all elements will have right inverses making End(A) into a group. But these is only one group of order 3, na ...
... Lemma 4. If End(A) is finite and f ∈ End(A) satisfies left cancellation then f has a right inverse (a morphism h : A → A so that f ◦ h = idA . If both f and g satisfies left cancellation then all elements will have right inverses making End(A) into a group. But these is only one group of order 3, na ...
Notes on categories - Math User Home Pages
... that is, we split S into disjoint subsets in all possible ways, equip those with the respective structures, then take the disjoint union of all those assignments. This looks like a convolution, and indeed the theory of species is supposed to categorify generating functions. Also note that we think o ...
... that is, we split S into disjoint subsets in all possible ways, equip those with the respective structures, then take the disjoint union of all those assignments. This looks like a convolution, and indeed the theory of species is supposed to categorify generating functions. Also note that we think o ...
Binary Structures
... Guidelines for proving the non-isomorphism of binary structures There exist only one way to prove that two binary structures S and S0 are not isomorphic: find a property that must be shared by any isomorphic structures (structural property) but distinguishes S and S0 . Example 9. Because any isomorp ...
... Guidelines for proving the non-isomorphism of binary structures There exist only one way to prove that two binary structures S and S0 are not isomorphic: find a property that must be shared by any isomorphic structures (structural property) but distinguishes S and S0 . Example 9. Because any isomorp ...
NATURAL TRANSFORMATIONS Id −→ Id Here is a categorical way
... can annihilate all of the quotient modules R/(rs). This generalizes Mitya’s version of the counterexample that I chose to present in class by crude direct calculation. We’re getting more sophisticated now! Example 6. There are no non-identity natural transformations of the identity functor on the ca ...
... can annihilate all of the quotient modules R/(rs). This generalizes Mitya’s version of the counterexample that I chose to present in class by crude direct calculation. We’re getting more sophisticated now! Example 6. There are no non-identity natural transformations of the identity functor on the ca ...
PPT
... the same parity of the number of transpositions Definition: A permutation is even if it can be factored into an even number of transpositions A permutation is odd if it can be factored into an odd number of transpositions ...
... the same parity of the number of transpositions Definition: A permutation is even if it can be factored into an even number of transpositions A permutation is odd if it can be factored into an odd number of transpositions ...
MATH 6280 - CLASS 2 Contents 1. Categories 1 2. Functors 2 3
... (8) The category associated to a poset P with object elements p ∈ P and one morphisms p → q if p ≤ q. (9) The category VectF of vector spaces with morphisms linear transformations. (10) There is a canonical way to make a monoid M into a category BM with one object • where HomBM (•, •) = M . (11) To ...
... (8) The category associated to a poset P with object elements p ∈ P and one morphisms p → q if p ≤ q. (9) The category VectF of vector spaces with morphisms linear transformations. (10) There is a canonical way to make a monoid M into a category BM with one object • where HomBM (•, •) = M . (11) To ...
Jan Bergstra
... We refer to [1], [2] for such notions as functional and relational effective definitional scheme (feds and reds,respectively), and for the semantics of the logic of effective deffinitions. If a is an algebraic structure then Gct denotes the signature of a , we always assume that a contains only a fi ...
... We refer to [1], [2] for such notions as functional and relational effective definitional scheme (feds and reds,respectively), and for the semantics of the logic of effective deffinitions. If a is an algebraic structure then Gct denotes the signature of a , we always assume that a contains only a fi ...
Category Theory Example Sheet 1
... for all a, b, c ∈ L). Show that there is a category MatL whose objects are the natural numbers, and whose morphisms n −→ m are m × n matrices with entries from L, where we define ‘multiplication’ of such matrices by analogy with that of matrices over a field, interpreting ∧ as multiplication and ∨ a ...
... for all a, b, c ∈ L). Show that there is a category MatL whose objects are the natural numbers, and whose morphisms n −→ m are m × n matrices with entries from L, where we define ‘multiplication’ of such matrices by analogy with that of matrices over a field, interpreting ∧ as multiplication and ∨ a ...
Combinatorial species
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced by André Joyal.The power of the theory comes from its level of abstraction. The ""description format"" of a structure (such as adjacency list versus adjacency matrix for graphs) is irrelevant, because species are purely algebraic. Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.