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Showing content from https://en.cppreference.com/w/cpp/language/../error/error_code/../../locale/locale/operator().html below:

std::locale::operator() - cppreference.com

template< class CharT, class Traits, class Alloc >

bool operator()( const std::basic_string<CharT,Traits,Alloc>& s1,

                 const std::basic_string<CharT,Traits,Alloc>& s2 ) const;

Compares two string arguments s1 and s2 according to the lexicographic comparison rules defined by this locale's std::collate<CharT> facet. This operator allows any locale object that has a collate facet to be used as a binary predicate in the standard algorithms (such as std::sort) and ordered containers (such as std::set).

[edit] Parameters s1 - the first string to compare s2 - the second string to compare [edit] Return value

true if s1 is lexicographically less than s2, false otherwise.

[edit] Possible implementation 
template<class CharT, class Traits, class Alloc>
bool operator()(const std::basic_string<CharT,Traits,Alloc>& s1,
                const std::basic_string<CharT,Traits,Alloc>& s2) const
{
    return std::use_facet<std::collate<CharT>>(*this).compare(
               s1.data(), s1.data() + s1.size(),
               s2.data(), s2.data() + s2.size()) < 0;
}
[edit] Example

A vector of strings can be sorted according to a non-default locale by using the locale object as comparator:

#include <algorithm>
#include <cassert>
#include <locale>
#include <string>
#include <vector>
 
int main()
{
    std::vector<std::wstring> v = {L"жил", L"был", L"пёс"};
    std::sort(v.begin(), v.end(), std::locale("ru_RU.UTF8"));
    assert(v[0] == L"был");
    assert(v[1] == L"жил");
    assert(v[2] == L"пёс");
}
[edit] See also defines lexicographical comparison and hashing of strings
(class template) [edit]

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