#99 ⟨a, b, c | aa=b, ccc=1⟩

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. c3 ⇒ 1 [2]
2. b ⇒ a2 [1]
# abc:aa=b,ccc=1 ac/b - -
ccc=1
b=aa

Other submonoids of same group

2 unique, 30 total

Σ#PresentationPropertiesDescriptionφ
658⟨a, b, c | aaa=1, bcb=1⟩Grp Infℤ3 ∗ ℤ27
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.

25 total

Σ#PresentationMapping
6119⟨a, b, c | ab=c, aaa=1⟩φ(a) = c, φ(b) = a, φ(c) = ca
7280⟨a, b, c | aaa=b, ccc=1⟩φ(a) = a, φ(b) = aaa, φ(c) = c
7330⟨a, b, c | aab=c, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = aac
7376⟨a, b, c | aba=c, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = aca
7384⟨a, b, c | abb=c, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = acc
7425⟨a, b, c | bab=c, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = cac
7466⟨a, b, c | ba=ac, aaa=1⟩φ(a) = c, φ(b) = a, φ(c) = ccac
7488⟨a, b, c | bb=ac, aaa=1⟩φ(a) = c, φ(b) = a, φ(c) = ccaa
81589⟨a, b, c | aaa=bc, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ccaaa
81706⟨a, b, c | aab=bc, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ccaac
81756⟨a, b, c | aab=cb, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = aa
81870⟨a, b, c | aba=bc, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ccaca
81905⟨a, b, c | abb=bc, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ccacc
81907⟨a, b, c | abb=cb, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
81908⟨a, b, c | abb=cc, bbb=1⟩φ(a) = aac, φ(b) = c, φ(c) = a
82030⟨a, b, c | bab=bc, bbb=1⟩φ(a) = a, φ(b) = c, φ(c) = ac
82031⟨a, b, c | bab=cc, bbb=1⟩φ(a) = ccaacc, φ(b) = c, φ(c) = a
84106⟨a, b, c | aaa=1, aaba=c⟩φ(a) = c, φ(b) = a, φ(c) = ccac
84107⟨a, b, c | aaa=1, aabb=c⟩φ(a) = c, φ(b) = a, φ(c) = ccaa
84111⟨a, b, c | aaa=1, abab=c⟩φ(a) = c, φ(b) = a, φ(c) = caca
84115⟨a, b, c | aaa=1, abba=c⟩φ(a) = c, φ(b) = a, φ(c) = caac
84116⟨a, b, c | aaa=1, abbb=c⟩φ(a) = c, φ(b) = a, φ(c) = caaa
84128⟨a, b, c | aaa=1, baab=c⟩φ(a) = c, φ(b) = a, φ(c) = acca
84131⟨a, b, c | aaa=1, babb=c⟩φ(a) = c, φ(b) = a, φ(c) = acaa
84141⟨a, b, c | aaa=1, bbbb=c⟩φ(a) = c, φ(b) = a, φ(c) = aaaa