The Morphisms of Influence (Actions and Representations). More...

#include <algorithm>
#include <concepts>
#include <cstddef>
#include <functional>
#include <ranges>
#include <type_traits>
#include <utility>
import dedekind.category:species;

Include dependency graph for action.cppm:

Classes
struct	dedekind::category::is_linear_action< M, S, AddM, AddS, MultS, Act >
	The semimodule laws of an action `Act` : S x M -> M, declared: 1 m = m, 0 m = 0, (s + t) m = s m + t m, (s t) m = s (t m) and s (m + n) = s m + s n. A certificate, false where none is declared: a carrier with the right signatures is not thereby a semimodule. For scalars of type `size_t`, the integers modulo 2^64, the laws make n m the n-fold sum of m for every n. More...
struct	dedekind::category::is_linear_action< S, S, Add, Add, Mult, Mult >
	A commutative semiring over itself, by its own multiplication: the semimodule laws are the semiring's (Golan, Semirings and their Applications, 1999, ch. 14), among them the unsigned integers, the ring of integers modulo a power of two. More...
struct	dedekind::category::is_linear_action< S, S, Add, Add, std::multiplies< S >, std::multiplies<> >
	The same, acted on by the transparent `std::multiplies<>`. More...
struct	dedekind::category::counting_action< T, Op >
	The counts' action on a monoid (T, Op), \((n, x) \mapsto x^n\), the `n` copies of `x`, where its laws give it in one step: derived, never declared, and absent where no law gives it, so that the binary method of :total squares instead. More...
struct	dedekind::category::counting_action< T, Op >
	By the semimodule laws over the counts, \(1 \cdot x = x\) and \((s + t) \cdot x = s \cdot x + t \cdot x\), \(x^n\) is \(n \cdot x\), one action. More...
struct	dedekind::category::ZeroAction< A, B, Op >
	The 'Absorption' logic. More...
struct	dedekind::category::acting_monoid< H >
	A fold along the walk, `fold(r, v, h)`, and its specialisation by the action law (the Jlt paper's Listing 4). `acting_monoid<H>` names the monoid an action `h` acts by on the right, a declared certificate of \(h(h(v, a), b) = h(v, a \cdot b)\) and \(h(v, 1) = v\), the product in the walk's order; keyed by the action's type without cv-qualification, as a fold deduces it. A fold walks a right action; a left action, scalar first as `IsLeftAction`, is a right action of the opposite monoid and registers `Opposite<Op>` (:involution), the bridge between the two textbook conventions. On `n` copies of one value the fold is `v` acted on by their product, and `concat` computes the product in one step or by squaring: two declared laws, two overloads, composed by resolution. More...
struct	dedekind::category::acting_monoid< std::plus< std::size_t > >
	A monoid acts on itself: (std::size_t, +) by `+`. More...

Namespaces
namespace	dedekind::category

Concepts
concept	dedekind::category::IsLeftAction
	The shape of a left action (S ⟳ M), \(S \times M \to M\), scalar first, as a module's: its laws are the certificate a concept above it reads (`is_linear_action` for a semimodule). A left action of a monoid \(M\) is a right action (`IsRightAction`) of its opposite, `Opposite<Op>` (:involution), and over a commutative monoid the two coincide.
concept	dedekind::category::detail_action::MultipliesInPlace
	A carrier whose product stays in it: no integer narrower than `int`, whose product promotes to a signed `int` that may overflow.
concept	dedekind::category::IsSemimodule
	A commutative monoid `M` on which the semiring `S` acts linearly.
concept	dedekind::category::HasCountingAction
	The counts act on (T, Op) in one step, by a law: the semimodule laws or idempotence.
concept	dedekind::category::IsRightAction
	`H` is a right action \(M \times S \to M\) of the monoid (S, Op) that `acting_monoid<H>` names: `M` is a right `S-set` (Mac Lane and Moerdijk), an `S-act` (Kilp, Knauer and Mikhalev, Monoids, Acts and Categories). A left action is a right action of the opposite, `Opposite<Op>`.

Typedefs
using	dedekind::category::ZeroZ = decltype(zero<int, int, std::plus<int>>())

Functions
template<std::ranges::input_range R, typename T = std::ranges::range_value_t<R>, typename Op = std::plus<T>> requires IsMonoid<T, Op> && IsCopies<R> && HasCountingAction<T, Op>
constexpr T	dedekind::category::concat (R r, Op op={})
	`concat` on `n` copies of one value where the counts act in one step: \(x^n\) by the counting action, \(n \cdot x\) over a semimodule, `x` under an idempotent operation. The same signature as the binary method of :total and a stronger constraint, so resolution prefers it.
template<std::ranges::input_range R, std::semiregular M, typename H> requires std::invocable<const H&, M, std::ranges::range_value_t<R>>
constexpr M	dedekind::category::fold (R r, M v, H h)
	The fold along the walk: the standard's `fold_left`.
template<std::ranges::input_range R, std::semiregular M, typename H> requires IsCopies<R> && IsRightAction<H, M, std::ranges::range_value_t<R>>
constexpr M	dedekind::category::fold (R r, M v, H h)
	The fold under a monoid action of a sequence whose product has a fast form: `v` acted on once, by the product in the walk's order. The action law \(h(h(v, a), b) = h(v, a \cdot b)\) licenses this on any sequence, but it pays only where the product is cheaper than the actions one by one: elsewhere a product each costs more than an action each, a matrix by a matrix against a vector by a matrix. The fast form today is copies, whose product `concat` computes by squaring; periodic sequences are the natural next one. The same signature as the plain fold and a stronger constraint, so resolution prefers it.

Variables
template<typename M, typename S, typename AddM, typename AddS, typename MultS, typename Act>
constexpr bool	dedekind::category::is_linear_action_v
	Value form of `is_linear_action`.
constexpr std::size_t	dedekind::category::detail_fold_witness::ten = 10

Detailed Description

The Morphisms of Influence (Actions and Representations).

@partition :action

action__Structural_Influence

While Level 0 (:morphism) defines mapping between species, Level 1 defines the "Action" of a Species S upon a Species M (S ⟳ M). In the Dedekind structuralist view, an Action is the reification of Scalar multiplication, leading naturally to the concepts of Modules, Vector Spaces, and eventually, the Linear Algebra of the Continuum.

action__Declared_Laws

A signature is no law: the semimodule laws of an action are a declared certificate, is_linear_action, false where none is declared (#1099).

Note: « Качественные методы позволяют исследовать структуру пространства через инвариантные множества и их преобразования. » (Qualitative methods allow the study of the structure of space through invariant sets and their transformations.) — V. V. Stepanov (В. В. Степанов), 'Курс дифференциальных уравнений'

Classes

Namespaces

Concepts

Typedefs

Functions

Variables

Detailed Description

action__Structural_Influence

action__Declared_Laws