#253 ⟨a, b | aa=b, abb=a⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aa2a3a4
11aa2a3a4
aaa2a3a4a
a2a2a3a4aa2
a3a3a4aa2a3
a4a4aa2a3a4

Right 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. a5 ⇒ a [2]
2. b ⇒ a2 [1]
# ab:aa=b,abb=a a/b - -
aaaaa=a
b=aa

Same cardinality

11 unique, 1391 total

Σ#PresentationPropertiesDescriptionφ
644⟨a, b | aa=b, abb=1⟩Grp Com 5ℤ51132
7254⟨a, b | aa=b, abb=b⟩Noc Com 5ℕ(5 = 2)43
7268⟨a, b | ab=a, baa=b⟩Noc 586
8574⟨a, b | aaa=b, aab=b⟩Noc Com 5ℕ(5 = 3)27
8950⟨a, b | ab=a, bbbb=a⟩Noc Com 5ℕ(5 = 4)32
8995⟨a, b | ab=a, aaa=bb⟩Noc Com 519
81019⟨a, b | ab=a, bba=bb⟩Noc 525
81020⟨a, b | ab=a, bbb=aa⟩Noc Com 59
81022⟨a, b | ab=a, bbb=ba⟩Noc 54
108617⟨a, b | aa=a, abbbba=b⟩Noc Com 53
1115426⟨a, b | aaa=aa, abbba=b⟩Noc Com 5

Isomorphic instances

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

71 total

Σ#PresentationMapping
7255⟨a, b | aa=b, bab=a⟩φ(a) = a, φ(b) = aa
8573⟨a, b | aaa=b, aab=a⟩φ(a) = aaa, φ(b) = a
8575⟨a, b | aaa=b, aba=a⟩φ(a) = aaa, φ(b) = a
8902⟨a, b | aa=b, aaab=a⟩φ(a) = a, φ(b) = aa
8904⟨a, b | aa=b, aaba=a⟩φ(a) = a, φ(b) = aa
8921⟨a, b | ab=a, aaaa=b⟩φ(a) = a, φ(b) = aaaa
92058⟨a, b | aab=b, aabb=a⟩φ(a) = aa, φ(b) = a
92062⟨a, b | aab=b, abab=a⟩φ(a) = aa, φ(b) = a
92070⟨a, b | aab=b, baab=a⟩φ(a) = aa, φ(b) = a
92110⟨a, b | aba=b, aabb=a⟩φ(a) = aa, φ(b) = a
92112⟨a, b | aba=b, abab=a⟩φ(a) = aa, φ(b) = a
92114⟨a, b | aba=b, abba=a⟩φ(a) = aa, φ(b) = a
92896⟨a, b | aa=b, aaaaa=a⟩φ(a) = a, φ(b) = aa
92939⟨a, b | ab=a, aaaab=b⟩φ(a) = a, φ(b) = aaaa
92941⟨a, b | ab=a, aaaba=b⟩φ(a) = a, φ(b) = aaaa
92945⟨a, b | ab=a, aabaa=b⟩φ(a) = a, φ(b) = aaaa
92953⟨a, b | ab=a, abaaa=b⟩φ(a) = a, φ(b) = aaaa
106278⟨a, b | aaa=b, aaaaa=a⟩φ(a) = aaa, φ(b) = a
106321⟨a, b | aab=a, aaaab=b⟩φ(a) = aaa, φ(b) = a
106323⟨a, b | aab=a, aaaba=b⟩φ(a) = aaa, φ(b) = a
106327⟨a, b | aab=a, aabaa=b⟩φ(a) = aaa, φ(b) = a
106335⟨a, b | aab=a, abaaa=b⟩φ(a) = aaa, φ(b) = a
106451⟨a, b | aba=a, aaaba=b⟩φ(a) = aaa, φ(b) = a
106455⟨a, b | aba=a, aabaa=b⟩φ(a) = aaa, φ(b) = a
108719⟨a, b | ab=a, aaaabb=b⟩φ(a) = a, φ(b) = aaaa
108723⟨a, b | ab=a, aaabab=b⟩φ(a) = a, φ(b) = aaaa
108725⟨a, b | ab=a, aaabba=b⟩φ(a) = a, φ(b) = aaaa
108731⟨a, b | ab=a, aabaab=b⟩φ(a) = a, φ(b) = aaaa
108733⟨a, b | ab=a, aababa=b⟩φ(a) = a, φ(b) = aaaa
108737⟨a, b | ab=a, aabbaa=b⟩φ(a) = a, φ(b) = aaaa
108747⟨a, b | ab=a, abaaab=b⟩φ(a) = a, φ(b) = aaaa
108749⟨a, b | ab=a, abaaba=b⟩φ(a) = a, φ(b) = aaaa
108753⟨a, b | ab=a, ababaa=b⟩φ(a) = a, φ(b) = aaaa
108761⟨a, b | ab=a, abbaaa=b⟩φ(a) = a, φ(b) = aaaa
1114369⟨a, b | aaaa=b, aaaaa=a⟩φ(a) = a, φ(b) = aaaa
1114412⟨a, b | aaab=a, aaaab=b⟩φ(a) = a, φ(b) = aa
1114414⟨a, b | aaab=a, aaaba=b⟩φ(a) = a, φ(b) = aa
1114418⟨a, b | aaab=a, aabaa=b⟩φ(a) = a, φ(b) = aa
1114426⟨a, b | aaab=a, abaaa=b⟩φ(a) = a, φ(b) = aa
1114542⟨a, b | aaba=a, aaaba=b⟩φ(a) = a, φ(b) = aa
1114546⟨a, b | aaba=a, aabaa=b⟩φ(a) = a, φ(b) = aa
1114554⟨a, b | aaba=a, abaaa=b⟩φ(a) = a, φ(b) = aa
1118966⟨a, b | aab=b, aaaabb=a⟩φ(a) = aa, φ(b) = a
1118970⟨a, b | aab=b, aaabab=a⟩φ(a) = aa, φ(b) = a
1118978⟨a, b | aab=b, aabaab=a⟩φ(a) = aa, φ(b) = a
1118994⟨a, b | aab=b, abaaab=a⟩φ(a) = aa, φ(b) = a
1119026⟨a, b | aab=b, baaaab=a⟩φ(a) = aa, φ(b) = a
1119170⟨a, b | aba=b, aaabab=a⟩φ(a) = aa, φ(b) = a
1119176⟨a, b | aba=b, aabaab=a⟩φ(a) = aa, φ(b) = a
1119178⟨a, b | aba=b, aababa=a⟩φ(a) = aa, φ(b) = a
1119192⟨a, b | aba=b, abaaba=a⟩φ(a) = aa, φ(b) = a
1124347⟨a, b | ab=a, aaaabbb=b⟩φ(a) = a, φ(b) = aaaa
1124355⟨a, b | ab=a, aaababb=b⟩φ(a) = a, φ(b) = aaaa
1124359⟨a, b | ab=a, aaabbab=b⟩φ(a) = a, φ(b) = aaaa
1124361⟨a, b | ab=a, aaabbba=b⟩φ(a) = a, φ(b) = aaaa
1124371⟨a, b | ab=a, aabaabb=b⟩φ(a) = a, φ(b) = aaaa
1124375⟨a, b | ab=a, aababab=b⟩φ(a) = a, φ(b) = aaaa
1124377⟨a, b | ab=a, aababba=b⟩φ(a) = a, φ(b) = aaaa
1124383⟨a, b | ab=a, aabbaab=b⟩φ(a) = a, φ(b) = aaaa
1124385⟨a, b | ab=a, aabbaba=b⟩φ(a) = a, φ(b) = aaaa
1124389⟨a, b | ab=a, aabbbaa=b⟩φ(a) = a, φ(b) = aaaa
1124403⟨a, b | ab=a, abaaabb=b⟩φ(a) = a, φ(b) = aaaa
1124407⟨a, b | ab=a, abaabab=b⟩φ(a) = a, φ(b) = aaaa
1124409⟨a, b | ab=a, abaabba=b⟩φ(a) = a, φ(b) = aaaa
1124415⟨a, b | ab=a, ababaab=b⟩φ(a) = a, φ(b) = aaaa
1124417⟨a, b | ab=a, abababa=b⟩φ(a) = a, φ(b) = aaaa
1124421⟨a, b | ab=a, ababbaa=b⟩φ(a) = a, φ(b) = aaaa
1124431⟨a, b | ab=a, abbaaab=b⟩φ(a) = a, φ(b) = aaaa
1124433⟨a, b | ab=a, abbaaba=b⟩φ(a) = a, φ(b) = aaaa
1124437⟨a, b | ab=a, abbabaa=b⟩φ(a) = a, φ(b) = aaaa
1124445⟨a, b | ab=a, abbbaaa=b⟩φ(a) = a, φ(b) = aaaa