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Add FullTransformationMonoid and PartialTransformationMonoid pres…
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…entations (semigroups#371)

* Update relations

* Add relation that e_12 is idempotent (missing from book)

* Add test using presentation

* Add relations from full transformation monoid

* Perform error checking and formatting

* Increase degree for ToddCoxeter tests

* Fix errors

* Change test tags

* Add extreme test

* Improve lambda functions

* Perform formatting

* Fix compilation warning

* Add extreme test

* Intermediate progress: things work

* Sort author value

* Change Aizenstat to Sutov, for partial transformation monoid

* Move function to correct unnamed namespace

* Add suggestion to self

* Add comment for function

* Formatting

* Improve comment consistency

* Remove FIXME comment

* Fixup per James's review

* Change not for exclamation mark

* Fix bracket

* Perform formatting

* Update Sims1 test

Co-authored-by: Murray Whyte <[email protected]>
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@MTWhyte
MTWhyte and Murray Whyte authored Sep 28, 2022
1 parent badc1c4 commit 7b1aac1
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186 changes: 151 additions & 35 deletions tests/fpsemi-examples.cpp
View file
Original file line number Diff line number Diff line change
Expand Up @@ -99,6 +99,112 @@ namespace libsemigroups {
}
}
}
void add_iwahori_full_transformation_monoid_relations(
std::vector<relation_type>& result,
size_t n,
size_t m) {
// This function adds the full transformation monoid relations due to
// Iwahori, from Section 9.3, p161-162, (Ganyushkin + Mazorchuk),
// expressed in terms of the generating set {pi_2, ..., pi_n,
// epsilon_{12}} using the notation of that chapter.
// https://link.springer.com/book/10.1007/978-1-84800-281-4

// The argument m specifies the letter value the idempotent e12
// corresponds to. When adding these relations for the full transformation
// monoid presentation, we want m = n - 1. For the partial transformation
// monoid presentation, we want m = n.

// It is useful to have this as a separate function, to avoid computing
// duplicate presentations for the underlying symmetric group, when
// combining relations from the symmetric inverse and full transformation
// monoids.

word_type e12 = {m};
std::vector<word_type> pi;
for (size_t i = 0; i <= n - 2; ++i) {
pi.push_back({i});
}

// The following expresses the epsilon idempotents in terms of the
// generating set
auto eps = [&e12, &pi](size_t i, size_t j) -> word_type {
LIBSEMIGROUPS_ASSERT(i != j);
if (i == 1 && j == 2) {
return e12;
} else if (i == 2 && j == 1) {
return pi[0] * e12 * pi[0];
} else if (i == 1) {
return pi[0] * pi[j - 2] * pi[0] * e12 * pi[0] * pi[j - 2] * pi[0];
} else if (j == 2) {
return pi[i - 2] * e12 * pi[i - 2];
} else if (j == 1) {
return pi[0] * pi[i - 2] * e12 * pi[i - 2] * pi[0];
} else if (i == 2) {
return pi[j - 2] * pi[0] * e12 * pi[0] * pi[j - 2];
}
return pi[i - 2] * pi[0] * pi[j - 2] * pi[0] * e12 * pi[0] * pi[j - 2]
* pi[0] * pi[i - 2];
};

auto transp = [&pi](size_t i, size_t j) -> word_type {
LIBSEMIGROUPS_ASSERT(i != j);
if (i > j) {
std::swap(i, j);
}
if (i == 1) {
return pi[j - 2];
}
return pi[i - 2] * pi[j - 2] * pi[i - 2];
};

// Relations a
for (size_t i = 1; i <= n; ++i) {
for (size_t j = 1; j <= n; ++j) {
if (j == i) {
continue;
}
// Relations (k)
result.emplace_back(transp(i, j) * eps(i, j), eps(i, j));
// Relations (j)
result.emplace_back(eps(j, i) * eps(i, j), eps(i, j));
// Relations (i)
result.emplace_back(eps(i, j) * eps(i, j), eps(i, j));
// Relations (d)
result.emplace_back(transp(i, j) * eps(i, j) * transp(i, j),
eps(j, i));
for (size_t k = 1; k <= n; ++k) {
if (k == i || k == j) {
continue;
}
// Relations (h)
result.emplace_back(eps(k, j) * eps(i, j), eps(k, j));
// Relations (g)
result.emplace_back(eps(k, i) * eps(i, j),
transp(i, j) * eps(k, j));
// Relations (f)
result.emplace_back(eps(j, k) * eps(i, j), eps(i, j) * eps(i, k));
result.emplace_back(eps(j, k) * eps(i, j), eps(i, k) * eps(i, j));
// Relations (c)
result.emplace_back(transp(k, i) * eps(i, j) * transp(k, i),
eps(k, j));
// Relations (b)
result.emplace_back(transp(j, k) * eps(i, j) * transp(j, k),
eps(i, k));
for (size_t l = 1; l <= n; ++l) {
if (l == i || l == j || l == k) {
continue;
}
// Relations (e)
result.emplace_back(eps(l, k) * eps(i, j), eps(i, j) * eps(l, k));
// Relations (a)
result.emplace_back(transp(k, l) * eps(i, j) * transp(k, l),
eps(i, j));
}
}
}
}
}

} // namespace

std::vector<relation_type> RookMonoid(size_t l, int q) {
Expand Down Expand Up @@ -1092,34 +1198,45 @@ namespace libsemigroups {
return result;
}

// From Proposition 1.7 in https://bit.ly/3R5ZpKW
std::vector<relation_type> FullTransformationMonoidAizenstat(size_t n) {
std::vector<relation_type> FullTransformationMonoid(size_t n, author val) {
if (n < 4) {
LIBSEMIGROUPS_EXCEPTION(
"the 1st argument (size_t) must be at least 4, found %llu",
uint64_t(n));
}
auto result = SymmetricGroup1(n);
if (val == author::Aizenstat) {
// From Proposition 1.7 in https://bit.ly/3R5ZpKW
auto result = SymmetricGroup1(n);

word_type const a = {1};
word_type const b = {2};
word_type const t = {3};
word_type const a = {1};
word_type const b = {2};
word_type const t = {3};

result.emplace_back(a * t, t);
result.emplace_back(
(b ^ (n - 2)) * a * (b ^ 2) * t * (b ^ (n - 2)) * a * (b ^ 2), t);
result.emplace_back(b * a * (b ^ (n - 1)) * a * b * t * (b ^ (n - 1)) * a
* b * a * (b ^ (n - 1)),
t);
result.emplace_back((t * b * a * (b ^ (n - 1))) ^ 2, t);
result.emplace_back(((b ^ (n - 1)) * a * b * t) ^ 2,
t * (b ^ (n - 1)) * a * b * t);

result.emplace_back((t * (b ^ (n - 1)) * a * b) ^ 2,
t * (b ^ (n - 1)) * a * b * t);
result.emplace_back((t * b * a * (b ^ (n - 2)) * a * b) ^ 2,
(b * a * (b ^ (n - 2)) * a * b * t) ^ 2);
return result;
result.emplace_back(a * t, t);
result.emplace_back(
(b ^ (n - 2)) * a * (b ^ 2) * t * (b ^ (n - 2)) * a * (b ^ 2), t);
result.emplace_back(b * a * (b ^ (n - 1)) * a * b * t * (b ^ (n - 1)) * a
* b * a * (b ^ (n - 1)),
t);
result.emplace_back((t * b * a * (b ^ (n - 1))) ^ 2, t);
result.emplace_back(((b ^ (n - 1)) * a * b * t) ^ 2,
t * (b ^ (n - 1)) * a * b * t);

result.emplace_back((t * (b ^ (n - 1)) * a * b) ^ 2,
t * (b ^ (n - 1)) * a * b * t);
result.emplace_back((t * b * a * (b ^ (n - 2)) * a * b) ^ 2,
(b * a * (b ^ (n - 2)) * a * b * t) ^ 2);
return result;
} else if (val == author::Iwahori) {
// From Theorem 9.3.1, p161-162, (Ganyushkin + Mazorchuk)
// using Theorem 9.1.4 to express presentation in terms
// of the pi_i and e_12.
// https://link.springer.com/book/10.1007/978-1-84800-281-4
auto result = SymmetricGroup(n, author::Carmichael);
add_iwahori_full_transformation_monoid_relations(result, n, n - 1);
return result;
}
LIBSEMIGROUPS_EXCEPTION("not yet implemented");
}

std::vector<relation_type> PartialTransformationMonoid(size_t n, author val) {
Expand Down Expand Up @@ -1172,18 +1289,20 @@ namespace libsemigroups {
{{3, 1, 1, 2, 0}, {3, 1, 1, 2}},
{{3, 1, 1, 2, 1}, {3, 1, 0, 2}},
{{1, 2, 1, 2, 0, 2}, {2, 1, 2, 0, 2}}};
} else if (n >= 4 && val == author::Aizenstat) {
LIBSEMIGROUPS_EXCEPTION("not yet implemented");
// FIXME this is missing the relations from T_n
} else if (n >= 4 && val == author::Sutov) {
// From Theorem 9.4.1, p169, (Ganyushkin + Mazorchuk)
// https://link.springer.com/book/10.1007/978-1-84800-281-4
auto result = SymmetricInverseMonoid(n, author::Sutov);
word_type e12 = {2 * n - 1};
std::vector<word_type> epsilon = {{n - 1}, {n}, {n + 1}};
result.emplace_back(epsilon[1] * e12, e12);
result.emplace_back(e12 * epsilon[1], epsilon[1]);
result.emplace_back(epsilon[0] * e12, e12 * epsilon[0] * epsilon[1]);
result.emplace_back(epsilon[2] * e12, e12 * epsilon[2]);
auto result = SymmetricInverseMonoid(n, author::Sutov);

add_iwahori_full_transformation_monoid_relations(result, n, n);
word_type e12 = {n};
std::vector<word_type> epsilon = {{n - 1}, {0, n - 1, 0}, {1, n - 1, 1}};
result.emplace_back(e12 * epsilon[1], e12);
result.emplace_back(epsilon[1] * e12, epsilon[1]);

result.emplace_back(e12 * epsilon[0], epsilon[1] * epsilon[0] * e12);
result.emplace_back(e12 * epsilon[2], epsilon[2] * e12);

return result;
}
LIBSEMIGROUPS_EXCEPTION("not yet implemented");
Expand Down Expand Up @@ -1218,7 +1337,6 @@ namespace libsemigroups {
result.emplace_back(epsilon[1] * pi[k], pi[k] * epsilon[1]);
result.emplace_back(epsilon[k + 1] * pi[0], pi[0] * epsilon[k + 1]);
}

result.emplace_back(epsilon[1] * epsilon[0] * pi[0],
epsilon[1] * epsilon[0]);
return result;
Expand All @@ -1227,11 +1345,9 @@ namespace libsemigroups {
}

// Chinese monoid
// See: The Chinese Monoid - Cassaigne, Espie, Krob, Novelli and Hivert,
// 2001
// See: The Chinese Monoid - Cassaigne, Espie, Krob, Novelli and Hivert, 2001
std::vector<relation_type> ChineseMonoid(size_t n) {
std::vector<relation_type> result;

for (size_t a = 0; a < n; a++) {
for (size_t b = a; b < n; b++) {
for (size_t c = b; c < n; c++) {
Expand Down
12 changes: 10 additions & 2 deletions tests/fpsemi-examples.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,15 @@
#include "libsemigroups/types.hpp" // for relation_type

namespace libsemigroups {
enum class author { Machine, Aizenstat, Carmichael, East, Moore, Sutov };
enum class author {
Machine,
Aizenstat,
Carmichael,
East,
Iwahori,
Moore,
Sutov
};

std::vector<relation_type> RookMonoid(size_t l, int q);

Expand All @@ -56,7 +64,7 @@ namespace libsemigroups {
std::vector<relation_type> TemperleyLieb(size_t n);
std::vector<relation_type> Brauer(size_t n);
std::vector<relation_type> RectangularBand(size_t m, size_t n);
std::vector<relation_type> FullTransformationMonoidAizenstat(size_t n);
std::vector<relation_type> FullTransformationMonoid(size_t n, author val);
std::vector<relation_type> PartialTransformationMonoid(size_t n, author val);
std::vector<relation_type> SymmetricInverseMonoid(size_t n, author val);

Expand Down
87 changes: 87 additions & 0 deletions tests/test-knuth-bendix-6.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -437,5 +437,92 @@ namespace libsemigroups {
{{2, 2, 1}, {2, 1, 2}}}));
REQUIRE(kb.knuth_bendix().number_of_normal_forms(0, 10) == 1175);
}

LIBSEMIGROUPS_TEST_CASE("KnuthBendix",
"083",
"(cong) PartialTransformationMonoid4",
"[standard][congruence][knuth-bendix][cong]") {
auto rg = ReportGuard(REPORT);

size_t n = 4;
auto s = PartialTransformationMonoid(n, author::Sutov);
for (auto& rel : s) {
if (rel.first.empty()) {
rel.first = {n + 1};
}
if (rel.second.empty()) {
rel.second = {n + 1};
}
}
auto p = make<Presentation<word_type>>(s);
p.alphabet(n + 2);
presentation::add_identity_rules(p, n + 1);

KnuthBendix kb;
kb.set_number_of_generators(n + 2);
for (size_t i = 0; i < p.rules.size() - 1; i += 2) {
kb.add_pair(p.rules[i], p.rules[i + 1]);
}
REQUIRE(!kb.is_quotient_obviously_infinite());
REQUIRE(kb.number_of_classes() == 625);
}

LIBSEMIGROUPS_TEST_CASE("KnuthBendix",
"118",
"(cong) PartialTransformationMonoid5",
"[extreme][congruence][knuth-bendix][cong]") {
auto rg = ReportGuard(REPORT);

size_t n = 5;
auto s = PartialTransformationMonoid(n, author::Sutov);
for (auto& rel : s) {
if (rel.first.empty()) {
rel.first = {n + 1};
}
if (rel.second.empty()) {
rel.second = {n + 1};
}
}
auto p = make<Presentation<word_type>>(s);
p.alphabet(n + 2);
presentation::add_identity_rules(p, n + 1);

KnuthBendix kb;
kb.set_number_of_generators(n + 2);
for (size_t i = 0; i < p.rules.size() - 1; i += 2) {
kb.add_pair(p.rules[i], p.rules[i + 1]);
}
REQUIRE(!kb.is_quotient_obviously_infinite());
REQUIRE(kb.number_of_classes() == 7776);
}

LIBSEMIGROUPS_TEST_CASE("KnuthBendix",
"119",
"(cong) FullTransformationMonoid Iwahori",
"[extreme][congruence][knuth-bendix][cong]") {
auto rg = ReportGuard(REPORT);

size_t n = 5;
auto s = FullTransformationMonoid(n, author::Iwahori);
for (auto& rel : s) {
if (rel.first.empty()) {
rel.first = {n};
}
if (rel.second.empty()) {
rel.second = {n};
}
}
auto p = make<Presentation<word_type>>(s);
p.alphabet(n + 1);
presentation::add_identity_rules(p, n);
KnuthBendix kb;
kb.set_number_of_generators(n + 1);
for (size_t i = 0; i < p.rules.size() - 1; i += 2) {
kb.add_pair(p.rules[i], p.rules[i + 1]);
}
REQUIRE(!kb.is_quotient_obviously_infinite());
REQUIRE(kb.number_of_classes() == 3125);
}

} // namespace congruence
} // namespace libsemigroups
4 changes: 2 additions & 2 deletions tests/test-sims1.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -321,8 +321,8 @@ namespace libsemigroups {
REQUIRE(S.size() == 256);
auto p = make<Presentation<word_type>>(S);
REQUIRE(p.rules.size() == 132);
auto q
= make<Presentation<word_type>>(FullTransformationMonoidAizenstat(4));
auto q = make<Presentation<word_type>>(
FullTransformationMonoid(4, author::Aizenstat));
// q.rules.insert(q.rules.end(), p.rules.cbegin(), p.rules.cbegin() + 100);
// Using this presentation makes this perform terribly slowly.

Expand Down
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