monoid_concept.h

// Description: Monoid concept
// Documentation: algebra.txt

#ifndef PASTELSYS_MONOID_CONCEPT_H
#define PASTELSYS_MONOID_CONCEPT_H

#include "pastel/sys/algebra/semigroup_concept.h"

#include "pastel/sys/algebra/native_monoid.h"
#include "pastel/sys/ensure.h"

namespace Pastel
{

    //! An additive monoid.
    /*!
   A monoid (X, +, 0), where 0 in X, is a semi-group (X, +), 
   such that there exist 0 in X such that 
       
           x + 0 = x = 0 + x, for all x in X.
   */
    template <typename T>
    concept Additive_Monoid_Concept_ =
        Additive_SemiGroup_Concept<T> && 
        requires(T t) {
            // Returns whether t == 0.
            {zero(t)} -> std::convertible_to<bool>;
    };

    template <typename T>
    concept Additive_Monoid_Concept =
        Additive_Monoid_Concept_<RemoveCvRef<T>>;

    //! A multiplicative monoid.
    /*!
   A monoid (X, *, 1), where 1 in X, is a semi-group (X, *), 
   such that there exist 1 in X such that 
       
           x * 1 = x = 1 * x, for all x in X.
   */
    template <typename T>
    concept Multiplicative_Monoid_Concept_ =
        Multiplicative_SemiGroup_Concept<T> && 
        requires(T t) {
            // Returns whether t == 1.
            {one(t)} -> std::convertible_to<bool>;
            //! Returns the power t^p, for p in NN^{>= 0}.
            {pow(t, (integer)0)} -> std::convertible_to<T>;
    };

    template <typename T>
    concept Multiplicative_Monoid_Concept =
        Multiplicative_Monoid_Concept_<RemoveCvRef<T>>;

}

namespace Pastel
{

    //! Computes x^p, for p in NN^{>= 0}.
    /*!
   The notation x^p means to multiply x with itself p times.
   */
    template <Multiplicative_Monoid_Concept T>
    T monoidPower(T x, integer p)
    {
        ENSURE_OP(p, >=, 0);

        if (p == 0)
        {
            return 1;
        }

        return semiGroupPower(std::move(x), p);
    }

}

#endif