diff --git a/CMakeLists.txt b/CMakeLists.txt
index f80fb7f..59b9ad6 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -59,4 +59,5 @@ target_link_libraries(all_tests PRIVATE
         tensor_tests
         utils_tests
         vector_tests
+        hall_set_tests
         )
\ No newline at end of file
diff --git a/LibAlgebraUnitTests/CMakeLists.txt b/LibAlgebraUnitTests/CMakeLists.txt
index f2bbfd2..e8355e6 100644
--- a/LibAlgebraUnitTests/CMakeLists.txt
+++ b/LibAlgebraUnitTests/CMakeLists.txt
@@ -9,6 +9,7 @@ target_link_libraries(reporter PUBLIC UnitTest++)
 add_subdirectory(algebra)
 add_subdirectory(lie)
 add_subdirectory(signature_tests)
+add_subdirectory(hall_set)
 add_subdirectory(tensor)
 add_subdirectory(utils)
 add_subdirectory(vector)
diff --git a/LibAlgebraUnitTests/hall_set/CMakeLists.txt b/LibAlgebraUnitTests/hall_set/CMakeLists.txt
new file mode 100644
index 0000000..a0953d4
--- /dev/null
+++ b/LibAlgebraUnitTests/hall_set/CMakeLists.txt
@@ -0,0 +1,11 @@
+
+
+add_library(hall_set_tests OBJECT
+        test_extension_of_functions.cpp
+        )
+
+target_link_libraries(hall_set_tests PUBLIC UnitTest++ Libalgebra::Libalgebra reporter)
+
+add_executable(hall_set_tests_exe run_tests.cpp)
+target_link_libraries(hall_set_tests_exe PRIVATE hall_set_tests)
+
diff --git a/LibAlgebraUnitTests/hall_set/run_tests.cpp b/LibAlgebraUnitTests/hall_set/run_tests.cpp
new file mode 100644
index 0000000..9fd14bb
--- /dev/null
+++ b/LibAlgebraUnitTests/hall_set/run_tests.cpp
@@ -0,0 +1,14 @@
+//
+// Created by sam on 02/02/2021.
+//
+
+#include <UnitTest++/UnitTest++.h>
+#include "../reporter.h"
+
+int main()
+{
+    reporter report;
+    UnitTest::TestRunner runner(report);
+
+    return runner.RunTestsIf(UnitTest::Test::GetTestList(), NULL, UnitTest::True(), 0);
+}
\ No newline at end of file
diff --git a/LibAlgebraUnitTests/hall_set/test_extension_of_functions.cpp b/LibAlgebraUnitTests/hall_set/test_extension_of_functions.cpp
new file mode 100644
index 0000000..6796b2f
--- /dev/null
+++ b/LibAlgebraUnitTests/hall_set/test_extension_of_functions.cpp
@@ -0,0 +1,146 @@
+//
+// Created by sam on 13/09/2021.
+//
+
+#include <UnitTest++/UnitTest++.h>
+
+#include <libalgebra/libalgebra.h>
+
+using namespace alg;
+
+
+std::string letter_func(LET l)
+{
+    return std::to_string(l);
+}
+
+std::string binary_op(const std::string &l, const std::string &r)
+{
+    return "[" + l + "," + r + "]";
+}
+
+
+SUITE(hall_set_function_extensions) {
+
+
+    struct fixture
+    {
+
+        typedef hall_basis<5> hall_set_t;
+        typedef typename hall_set_t::KEY KEY;
+
+        template <typename Function, typename BinOp, typename Tag>
+        using extended_function = hall_set_t::extended_function<Function, BinOp,Tag>;
+
+        hall_set_t hall_set;
+
+
+        extended_function<decltype(&letter_func), decltype(&binary_op), typename hall_set_t::no_caching_tag>
+            non_cached_key2str;
+
+        extended_function<decltype(&letter_func), decltype(&binary_op), typename hall_set_t::lazy_cache_tag<3>>
+            lazy_cached_key2str;
+
+        extended_function<decltype(&letter_func), decltype(&binary_op), typename hall_set_t::lookup_table_tag<2>>
+            table_cached_key2str;
+
+        extended_function<decltype(&letter_func), decltype(&binary_op), typename hall_set_t::lookup_table_tag<0>>
+            lazy_nomax_cached_key2str;
+
+        fixture() : hall_set(),
+            non_cached_key2str(letter_func, binary_op, hall_set),
+            lazy_cached_key2str(letter_func, binary_op, hall_set),
+            table_cached_key2str(letter_func, binary_op, hall_set),
+            lazy_nomax_cached_key2str(letter_func, binary_op, hall_set)
+        {
+            hall_set.growup(5);
+        }
+
+
+
+    };
+
+
+    TEST_FIXTURE(fixture, test_letter_to_string) {
+        LET l = 1;
+
+        CHECK_EQUAL("1", hall_set.key2string(l));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_2) {
+        KEY k = 6;
+        CHECK_EQUAL("[1,2]", hall_set.key2string(k));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_3) {
+        KEY k = 16;
+        CHECK_EQUAL("[1,[1,2]]", hall_set.key2string(k));
+    }
+
+    TEST_FIXTURE(fixture, test_letter_to_string_non_cache) {
+        LET l = 1;
+
+        CHECK_EQUAL("1", non_cached_key2str(l));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_2_non_cache) {
+        KEY k = 6;
+        CHECK_EQUAL("[1,2]", non_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_3_non_cache) {
+        KEY k = 16;
+        CHECK_EQUAL("[1,[1,2]]", non_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_letter_to_string_lazy_cache) {
+        LET l = 1;
+
+        CHECK_EQUAL("1", lazy_cached_key2str(l));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_2_lazy_cache) {
+        KEY k = 6;
+        CHECK_EQUAL("[1,2]", lazy_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_3_lazy_cache) {
+        KEY k = 16;
+        CHECK_EQUAL("[1,[1,2]]", lazy_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_letter_to_string_table_cache) {
+        LET l = 1;
+
+        CHECK_EQUAL("1", table_cached_key2str(l));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_2_table_cache) {
+        KEY k = 6;
+        CHECK_EQUAL("[1,2]", table_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_3_table_cache) {
+        KEY k = 16;
+        CHECK_EQUAL("[1,[1,2]]", table_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_letter_to_string_lazy_nomax_cache) {
+        LET l = 1;
+
+        CHECK_EQUAL("1", lazy_nomax_cached_key2str(l));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_2_lazy_nomax_cache) {
+        KEY k = 6;
+        CHECK_EQUAL("[1,2]", lazy_nomax_cached_key2str(k));
+    }
+
+    TEST_FIXTURE(fixture, test_key_to_string_deg_3_lazy_nomax_cache) {
+        KEY k = 16;
+        CHECK_EQUAL("[1,[1,2]]", lazy_nomax_cached_key2str(k));
+    }
+
+
+}
+
diff --git a/libalgebra b/libalgebra
index c4baf20..28a68eb 160000
--- a/libalgebra
+++ b/libalgebra
@@ -1 +1 @@
-Subproject commit c4baf205ea49a3a5a3c24dfc14871199fc1aac9b
+Subproject commit 28a68eb63840a0ac3e7468c6872b2fa8ac66a7d0