PDF
... were investigated and studied by PRUFER in 1924, BAER in 1929. Generalizing the notion of ternary ring introduced by Lister [10], Dutta and Kar [6] introduced the notion of ternary semiring. Ternary semiring arises naturally as follows, consider the ring of integers Z which plays a vital role in the ...
... were investigated and studied by PRUFER in 1924, BAER in 1929. Generalizing the notion of ternary ring introduced by Lister [10], Dutta and Kar [6] introduced the notion of ternary semiring. Ternary semiring arises naturally as follows, consider the ring of integers Z which plays a vital role in the ...
Contents 1. Introduction 2 2. The monoidal background 5 2.1
... Our main result, theorem 4.3.1, is an acyclic models theorem for monoidal functors from a monoidal category C to the monoidal category C∗ (Z). We likewise establish several variations of this result, which cover the symmetric monoidal and the contravariant monoidal settings. As a consequence of our ...
... Our main result, theorem 4.3.1, is an acyclic models theorem for monoidal functors from a monoidal category C to the monoidal category C∗ (Z). We likewise establish several variations of this result, which cover the symmetric monoidal and the contravariant monoidal settings. As a consequence of our ...
Constructing quantales and their modules from monoidal
... One might ask how closely the three main examples of quantales and their modules are related to each other. For example, is it necessa.ry to directly verify that the objects in question are in fact quantales or modules, or do they arise from a single construction? This question was partly answered ( ...
... One might ask how closely the three main examples of quantales and their modules are related to each other. For example, is it necessa.ry to directly verify that the objects in question are in fact quantales or modules, or do they arise from a single construction? This question was partly answered ( ...
arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni
... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...
... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...
EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1
... In other words, we have “norm maps” only for the objects of F 0 . Since F 0 is closed under subobjects, we have a collection of pairs of subgroups K ⊂ H such that H/K ⊂ F 0 and any X ∈ F 0 can be expressed as a disjoint union of copies of various H/K. These parameterize the norms that an F 0 -commut ...
... In other words, we have “norm maps” only for the objects of F 0 . Since F 0 is closed under subobjects, we have a collection of pairs of subgroups K ⊂ H such that H/K ⊂ F 0 and any X ∈ F 0 can be expressed as a disjoint union of copies of various H/K. These parameterize the norms that an F 0 -commut ...
Fibrations of Predicates and Bicategories of Relations
... whose fibres are preorders. In the syntactic case described above, these are the term models that record only the existence of a proof of one proposition from another. Our results apply in full generality. Similarly, the locally preordered versions of the bicategories described in (2) above are well ...
... whose fibres are preorders. In the syntactic case described above, these are the term models that record only the existence of a proof of one proposition from another. Our results apply in full generality. Similarly, the locally preordered versions of the bicategories described in (2) above are well ...
The periodic table of n-categories for low
... monoid with multiplication given by composition. In Section 3 we examine “doubly degenerate” bicategories, that is, bicategories with only one 0-cell and 1-cell. Now the 2-cells have two compositions on them— horizontal and vertical. So we might expect the 2-cells to form some sort of structure with ...
... monoid with multiplication given by composition. In Section 3 we examine “doubly degenerate” bicategories, that is, bicategories with only one 0-cell and 1-cell. Now the 2-cells have two compositions on them— horizontal and vertical. So we might expect the 2-cells to form some sort of structure with ...
slides
... Let R be a commutative ring with a unit. Let M be a finite decomposition monoid. Then one can define the R-coalgebra R (M) (free module with basis M) ...
... Let R be a commutative ring with a unit. Let M be a finite decomposition monoid. Then one can define the R-coalgebra R (M) (free module with basis M) ...
Aspects of categorical algebra in initialstructure categories
... the first part, can again be applied in INS-categories, presented to algebraic categories over INS-categories. In particular it is shown that this implies that adjointness of « algebraic» functors over L induces adjointness of « algebraic » functors over an INS-category K . Since furthermore togethe ...
... the first part, can again be applied in INS-categories, presented to algebraic categories over INS-categories. In particular it is shown that this implies that adjointness of « algebraic» functors over L induces adjointness of « algebraic » functors over an INS-category K . Since furthermore togethe ...
PARTIALIZATION OF CATEGORIES AND INVERSE BRAID
... group, defined in [Da] and studied in [Wa, Ru, Mc, FRR]. The categorical partialization is again directly applicable and produces a category, in which it is natural to call endomorphism monoids the inverse braid-permutation monoids. These monoids are new. We describe all constructions necessary for ...
... group, defined in [Da] and studied in [Wa, Ru, Mc, FRR]. The categorical partialization is again directly applicable and produces a category, in which it is natural to call endomorphism monoids the inverse braid-permutation monoids. These monoids are new. We describe all constructions necessary for ...
Labelled combinatorial classes Contents 1 Labelling Atoms
... The class of sets of k-components of B is denoted S ETk (B). Formally it is defined as a k-sequence counted modulo permutation of the components. That is S ETk (B) = S EQk (B)/R where R is an equivalence relation that identifies two sequences if one is simply a permutation of the components of the o ...
... The class of sets of k-components of B is denoted S ETk (B). Formally it is defined as a k-sequence counted modulo permutation of the components. That is S ETk (B) = S EQk (B)/R where R is an equivalence relation that identifies two sequences if one is simply a permutation of the components of the o ...
Spectra of Small Categories and Infinite Loop Space Machines
... of the family (see 2.20 below). The way to do this is by using some natural transformations between them which arise from the subdivision functors between the interval categories. Definition 2.3 Let α, β ∈ N with β ≥ α. A functor t: Iβ → Iα such that t (0) = 0 and t (β) = α will be called a subdivis ...
... of the family (see 2.20 below). The way to do this is by using some natural transformations between them which arise from the subdivision functors between the interval categories. Definition 2.3 Let α, β ∈ N with β ≥ α. A functor t: Iβ → Iα such that t (0) = 0 and t (β) = α will be called a subdivis ...
Turboveg for Windows - Synbiosys.alterra.nl).
... Every dialog window has a Help button. Use it if you do not know what to do. 1. Create a database if no database is available or if you want to start with an empty one. From the menu select Database | New. The only two entries to be filled in are Database name and Range for system numbers. An empty ...
... Every dialog window has a Help button. Use it if you do not know what to do. 1. Create a database if no database is available or if you want to start with an empty one. From the menu select Database | New. The only two entries to be filled in are Database name and Range for system numbers. An empty ...
MONADS AND ALGEBRAIC STRUCTURES Contents 1
... functors that is central to the overall picture. In fact, in the abstracted, purely categorical thread of this paper, our approach towards monads is much to view them as belonging to the theory of adjoint functors by considering them as “traces” of adjunctions. A truly rigorous treatment of category ...
... functors that is central to the overall picture. In fact, in the abstracted, purely categorical thread of this paper, our approach towards monads is much to view them as belonging to the theory of adjoint functors by considering them as “traces” of adjunctions. A truly rigorous treatment of category ...
Weights for Objects of Monoids
... Weighted limits and colimits provide a uniform way to define many interesting operations on 2-categories. It is known for more than 40 years [Law] that the Eilenberg-Moore object for a monad T in a 2-category K is a weighted limit on a diagram defined by the monad T in the suspension on the simplici ...
... Weighted limits and colimits provide a uniform way to define many interesting operations on 2-categories. It is known for more than 40 years [Law] that the Eilenberg-Moore object for a monad T in a 2-category K is a weighted limit on a diagram defined by the monad T in the suspension on the simplici ...
Weights for Objects of Monoids
... Weighted limits and colimits provide a uniform way to define many interesting operations on 2-categories. It is known for more than 40 years [Law] that the Eilenberg-Moore object for a monad T in a 2-category K is a weighted limit on a diagram defined by the monad T in the suspension on the simplici ...
... Weighted limits and colimits provide a uniform way to define many interesting operations on 2-categories. It is known for more than 40 years [Law] that the Eilenberg-Moore object for a monad T in a 2-category K is a weighted limit on a diagram defined by the monad T in the suspension on the simplici ...
categories - Andrew.cmu.edu
... We think of the composition g ◦ f as a sort of “product” of the functions f and g, and consider abstract “algebras” of the sort arising from collections of functions. A category is just such an “algebra,” consisting of objects A, B, C, . . . and arrows f : A → B, g : B → C, . . . , that are closed u ...
... We think of the composition g ◦ f as a sort of “product” of the functions f and g, and consider abstract “algebras” of the sort arising from collections of functions. A category is just such an “algebra,” consisting of objects A, B, C, . . . and arrows f : A → B, g : B → C, . . . , that are closed u ...
07_chapter 2
... F: A A form a semiring, where addition is pointwise addition and multiplication is function composition. The zero morphism and the identity are the respective neutral elements. If A is the additive monoid of natural numbers we obtain the semiring of natural numbers as End(A), and if A = S^n with S ...
... F: A A form a semiring, where addition is pointwise addition and multiplication is function composition. The zero morphism and the identity are the respective neutral elements. If A is the additive monoid of natural numbers we obtain the semiring of natural numbers as End(A), and if A = S^n with S ...
- Departament de matemàtiques
... monoid as well as the Eckmann-Hilton argument make sense in any monoidal category in place of the category of sets. A double monoid in Cat is the same thing as a category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to ...
... monoid as well as the Eckmann-Hilton argument make sense in any monoidal category in place of the category of sets. A double monoid in Cat is the same thing as a category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to ...
On finite congruence
... This semiring is additively idempotent yet has order 3 and is not of the form V (G). At present, we have no strongly supported conjecture for a meaningful description of the semirings in the fourth case of Theorem 4.1, though we do believe that some good description might be possible. ...
... This semiring is additively idempotent yet has order 3 and is not of the form V (G). At present, we have no strongly supported conjecture for a meaningful description of the semirings in the fourth case of Theorem 4.1, though we do believe that some good description might be possible. ...
Semantical evaluations as monadic second-order
... 1) The existence of properties forming an inductive set (w.r.t. operations of F) is equivalent to recognizability in the considered F-algebra. 2) The simultaneous computation of m inductive properties can be implemented by a "tree" automaton with 2m states working on terms t. This computation takes ...
... 1) The existence of properties forming an inductive set (w.r.t. operations of F) is equivalent to recognizability in the considered F-algebra. 2) The simultaneous computation of m inductive properties can be implemented by a "tree" automaton with 2m states working on terms t. This computation takes ...
Generating sets of finite singular transformation semigroups
... n − 1). It is clear that α ∈ Dn−1 if and only if there exist i, j ∈ Xn with i = j such that ker(α) is the equivalence relation on Xn generated by {(i, j )}, or equivalently, generated by {(j, i)}. In this case we define the set Ker(α) by Ker(α) = {i, j }. (Notice that ker(α) denotes an equivalence ...
... n − 1). It is clear that α ∈ Dn−1 if and only if there exist i, j ∈ Xn with i = j such that ker(α) is the equivalence relation on Xn generated by {(i, j )}, or equivalently, generated by {(j, i)}. In this case we define the set Ker(α) by Ker(α) = {i, j }. (Notice that ker(α) denotes an equivalence ...
Notes - Mathematics and Statistics
... (10) Let S be a monoid (a semi-group with a two sided inverse). That is S has an associative composition law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functio ...
... (10) Let S be a monoid (a semi-group with a two sided inverse). That is S has an associative composition law, (x, y ) 7→ xy , with a two-sided inverse. Define a category C , with a single object ∗ and with Mor(∗, ∗) = S, where composition is given by multiplication. (So morphisms need not be functio ...
The Functor Category in Relation to the Model Theory of Modules
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
... pp-formulas = finitely generated subfunctors of representables: If φ is a pp-formula, Fφ is a finitely generated subfunctor of a representable functor in fp(mod(R), Ab). If F is a finitely generated subfunctor of a representable functor, then there exists pp-formula φ such that F ∼ = Fφ . ...
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.