#2064 ⟨a, b | aab=b, abba=a⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1abbab2b2ab3b3ab4
11abbab2b2ab3b3ab4
aab2ab3b3ab4abbab2
bbbab2b2ab3b3ab4ab
babab3ab4abbab2b2ab3
b2b2b2ab3b3ab4abbab2
b2ab2aabbab2b2ab3b3ab4
b3b3b3ab4abbab2b2ab3
b3ab3abab2b2ab3b3ab4ab
b4b4abbab2b2ab3b3ab4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. b5 ⇒ b [8]
2. ab ⇒ b3 [7]
3. b4a ⇒ a [6]
4. a2 ⇒ b2a [3]
# ab:aab=b,abba=a b/a - -
bbbbb=b
ab=bbb
bbbba=a
aa=bba

Same cardinality

26 unique, 904 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

14 total

Σ#PresentationMapping
92072⟨a, b | aab=b, baba=a⟩φ(a) = a, φ(b) = b
92076⟨a, b | aab=b, bbaa=a⟩φ(a) = a, φ(b) = b
106449⟨a, b | aba=a, aaaab=b⟩φ(a) = ba, φ(b) = b
1114540⟨a, b | aaba=a, aaaab=b⟩φ(a) = b, φ(b) = a
1118972⟨a, b | aab=b, aaabba=a⟩φ(a) = a, φ(b) = b
1118980⟨a, b | aab=b, aababa=a⟩φ(a) = a, φ(b) = b
1118984⟨a, b | aab=b, aabbaa=a⟩φ(a) = a, φ(b) = b
1118996⟨a, b | aab=b, abaaba=a⟩φ(a) = a, φ(b) = b
1119000⟨a, b | aab=b, ababaa=a⟩φ(a) = a, φ(b) = b
1119008⟨a, b | aab=b, abbaaa=a⟩φ(a) = a, φ(b) = b
1119028⟨a, b | aab=b, baaaba=a⟩φ(a) = a, φ(b) = b
1119032⟨a, b | aab=b, baabaa=a⟩φ(a) = a, φ(b) = b
1119040⟨a, b | aab=b, babaaa=a⟩φ(a) = a, φ(b) = b
1119056⟨a, b | aab=b, bbaaaa=a⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

24 total

Σ#PresentationMapping
92969⟨a, b | ab=a, baaaa=b⟩φ(a) = b, φ(b) = bba
106351⟨a, b | aab=a, baaaa=b⟩φ(a) = ba, φ(b) = b
108779⟨a, b | ab=a, baaaab=b⟩φ(a) = b, φ(b) = bba
108781⟨a, b | ab=a, baaaba=b⟩φ(a) = b, φ(b) = bba
108785⟨a, b | ab=a, baabaa=b⟩φ(a) = b, φ(b) = bba
108793⟨a, b | ab=a, babaaa=b⟩φ(a) = b, φ(b) = bba
108809⟨a, b | ab=a, bbaaaa=b⟩φ(a) = b, φ(b) = bba
1114442⟨a, b | aaab=a, baaaa=b⟩φ(a) = b, φ(b) = a
1114570⟨a, b | aaba=a, baaaa=b⟩φ(a) = b, φ(b) = a
1124467⟨a, b | ab=a, baaaabb=b⟩φ(a) = b, φ(b) = bba
1124471⟨a, b | ab=a, baaabab=b⟩φ(a) = b, φ(b) = bba
1124473⟨a, b | ab=a, baaabba=b⟩φ(a) = b, φ(b) = bba
1124479⟨a, b | ab=a, baabaab=b⟩φ(a) = b, φ(b) = bba
1124481⟨a, b | ab=a, baababa=b⟩φ(a) = b, φ(b) = bba
1124485⟨a, b | ab=a, baabbaa=b⟩φ(a) = b, φ(b) = bba
1124495⟨a, b | ab=a, babaaab=b⟩φ(a) = b, φ(b) = bba
1124497⟨a, b | ab=a, babaaba=b⟩φ(a) = b, φ(b) = bba
1124501⟨a, b | ab=a, bababaa=b⟩φ(a) = b, φ(b) = bba
1124509⟨a, b | ab=a, babbaaa=b⟩φ(a) = b, φ(b) = bba
1124527⟨a, b | ab=a, bbaaaab=b⟩φ(a) = b, φ(b) = bba
1124529⟨a, b | ab=a, bbaaaba=b⟩φ(a) = b, φ(b) = bba
1124533⟨a, b | ab=a, bbaabaa=b⟩φ(a) = b, φ(b) = bba
1124541⟨a, b | ab=a, bbabaaa=b⟩φ(a) = b, φ(b) = bba
1124557⟨a, b | ab=a, bbbaaaa=b⟩φ(a) = b, φ(b) = bba