#58 ⟨a, b, c | aaa=1, bcb=1⟩

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Contents

  1. Properties
  2. Rewriting system
  3. Other submonoids of same group
  4. Isomorphic instances

Properties

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. cb ⇒ bc [3]
2. a3 ⇒ 1 [1]
3. b2c ⇒ 1 [4]
# abc:aaa=1,bcb=1 abc - -
cb=bc
aaa=1
bbc=1

Other submonoids of same group

2 unique, 28 total

Σ#PresentationPropertiesDescriptionφ
699⟨a, b, c | aa=b, ccc=1⟩Can Infℤ3 ∗ ℕ25
7445⟨a, b, c | ac=ab, aaa=1⟩Can Infℤ3 ∗ ℕ1

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

27 total

Σ#PresentationMapping
7383⟨a, b, c | aba=c, ccc=1⟩φ(a) = b, φ(b) = bcabc, φ(c) = a
7569⟨a, b, c | aaa=1, abcb=1⟩φ(a) = bcab, φ(b) = bc, φ(c) = aabb
7573⟨a, b, c | aaa=1, bacb=1⟩φ(a) = a, φ(b) = bc, φ(c) = aabb
7576⟨a, b, c | aaa=1, bbcb=1⟩φ(a) = a, φ(b) = b, φ(c) = bcbcbc
71185⟨a, b, c | ab=1, baccc=1⟩φ(a) = bc, φ(b) = b, φ(c) = a
71208⟨a, b, c | ab=1, bccca=1⟩φ(a) = bc, φ(b) = b, φ(c) = a
71219⟨a, b, c | ab=1, cbacc=1⟩φ(a) = bc, φ(b) = b, φ(c) = a
81903⟨a, b, c | aba=cc, ccc=1⟩φ(a) = b, φ(b) = bcaabc, φ(c) = a
83607⟨a, b, c | aaa=1, aabcb=1⟩φ(a) = a, φ(b) = b, φ(c) = bcaaaabc
83612⟨a, b, c | aaa=1, abacb=1⟩φ(a) = a, φ(b) = b, φ(c) = aabcaabc
83617⟨a, b, c | aaa=1, abbcb=1⟩φ(a) = bcbcabb, φ(b) = bc, φ(c) = aabbb
83619⟨a, b, c | aaa=1, abcab=1⟩φ(a) = a, φ(b) = b, φ(c) = bcaabcaa
83621⟨a, b, c | aaa=1, abcba=1⟩φ(a) = a, φ(b) = b, φ(c) = bcaaaabc
83622⟨a, b, c | aaa=1, abcbb=1⟩φ(a) = bcab, φ(b) = bc, φ(c) = aabbb
83628⟨a, b, c | aaa=1, baacb=1⟩φ(a) = a, φ(b) = b, φ(c) = aaaabcbc
83632⟨a, b, c | aaa=1, babcb=1⟩φ(a) = bcab, φ(b) = bc, φ(c) = aabbb
83634⟨a, b, c | aaa=1, bacab=1⟩φ(a) = a, φ(b) = b, φ(c) = aabcbcaa
83635⟨a, b, c | aaa=1, bacbb=1⟩φ(a) = a, φ(b) = bc, φ(c) = aabbb
83640⟨a, b, c | aaa=1, bbacb=1⟩φ(a) = a, φ(b) = bc, φ(c) = aabbb
83643⟨a, b, c | aaa=1, bbbcb=1⟩φ(a) = a, φ(b) = b, φ(c) = bcbcbcbc
83645⟨a, b, c | aaa=1, bbcbb=1⟩φ(a) = a, φ(b) = b, φ(c) = bcbcbcbc
84122⟨a, b, c | aaa=1, abcb=a⟩φ(a) = a, φ(b) = b, φ(c) = bcbc
84135⟨a, b, c | aaa=1, bacb=a⟩φ(a) = a, φ(b) = b, φ(c) = aabcabc
84145⟨a, b, c | aaa=1, bbcb=a⟩φ(a) = bcab, φ(b) = b, φ(c) = bcbcbca
84421⟨a, b, c | aba=1, accc=a⟩φ(a) = b, φ(b) = bcbc, φ(c) = a
84461⟨a, b, c | aba=1, bccc=b⟩φ(a) = b, φ(b) = bcbc, φ(c) = a
86557⟨a, b, c | ab=1, bcbcbc=1⟩φ(a) = bc, φ(b) = b, φ(c) = bca