#253 ⟨a, b | aa=b, abb=a⟩

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Quick links

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

Properties

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.

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 to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 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#PresentationDescriptionRelated
644⟨a, b | aa=b, abb=1⟩Isomorphic to ℤ51132 iso
7254⟨a, b | aa=b, abb=b⟩Isomorphic to ℕ(5 = 2)43 iso
7268⟨a, b | ab=a, baa=b⟩Finite non-commutative monoid with 5 elements63 iso, 23 anti-iso
8574⟨a, b | aaa=b, aab=b⟩Isomorphic to ℕ(5 = 3)27 iso
8950⟨a, b | ab=a, bbbb=a⟩Isomorphic to ℕ(5 = 4)32 iso
8995⟨a, b | ab=a, aaa=bb⟩Finite commutative monoid with 5 elements19 iso
81019⟨a, b | ab=a, bba=bb⟩Finite non-commutative monoid with 5 elements25 iso
81020⟨a, b | ab=a, bbb=aa⟩Finite commutative monoid with 5 elements9 iso
81022⟨a, b | ab=a, bbb=ba⟩Finite non-commutative monoid with 5 elements4 iso
108617⟨a, b | aa=a, abbbba=b⟩Finite commutative monoid with 5 elements3 iso
1115426⟨a, b | aaa=aa, abbba=b⟩Finite commutative monoid with 5 elements

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