Action

template<typename TElementType, typename TPointType, typename TActionType, typename TTraits, side TLeftOrRight>
class Action : public libsemigroups::Runner, private libsemigroups::detail::BruidhinnTraits<TPointType>

Defined in action.hpp.

This page contains details of the Action class in libsemigroups for finding actions of semigroups, or groups, on sets. The notion of an “action” in the context of libsemigroups is analogous to the notion of an orbit of a group.

You are unlikely to want to use this class directly directly, but rather via the more convenient aliases libsemigroups::RightAction and libsemigroups::LeftAction.

The

run member function finds points that can be obtained by acting on the seeds of this by the generators of this until no further points can be found, or Runner::stopped returns true. This is achieved by performing a breadth first search.
See

side, ActionTraits, LeftAction, and RightAction.

Example

using namespace libsemigroups;
using PPerm = PPermHelper<16>::type;
RightAction<PPerm, PPerm, ImageRightAction<PPerm, PPerm>> o;
o.add_seed(PPerm::identity(16));
o.add_generator(
    PPerm({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
          {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0},
          16));
o.add_generator(
    PPerm({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
          {1, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
          16));
o.add_generator(PPerm({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
                      {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14},
                      16));
o.add_generator(PPerm({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14},
                      {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
                      16));
o.reserve(70000);
o.size();              // 65536
o.digraph().nr_scc();  // 17

Template Parameters

  • TElementType: the type of the elements of the semigroup.

  • TPointType: the type of the points acted on.

  • TActionType: the type of the action of elements of type TElementType on points of type TPointType. This type should be a stateless trivially default constructible class with a call operator of signature void(TPointType&, TPointType const&, TElementType const&) which computes the action of the 3rd argument on the 2nd argument, and stores it in the 1st argument. See, for example, libsemigroups::ImageLeftAction and libsemigroups::ImageRightAction.

  • TTraits: the type of a traits class with the requirements of libsemigroups::ActionTraits.

  • TLeftOrRight: the libsemigroups::side of the action (i.e. if it is is a left or a right action).

Complexity

The time complexity is \(O(mn)\) where \(m\) is the total number of points in the orbit and \(n\) is the number of generators.