#12187 ⟨a, b | aaaa=ab, babb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 18 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a⁷ = a⁴ and a⁴ ⋅ 1 = a⁴, however a⁷ ≠ 1

Elements

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) = x^m.

1 element center:
- 1
2 non-trivial idempotents:
- ab², b⁷
Order of generators:
- a: index 4, period 7
- b: index 1, period 7
Histogram:

index 1, period 1 2 elements ab², b⁷

index 1, period 7 12 elements b, ab, b², b³, ab³, ...

index 2, period 7 2 elements a², a³

index 4, period 7 1 element a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a³	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷
1	1	a	b	a²	ab	b²	a³	ab²	b³	ab³	b⁴	ab⁴	b⁵	ab⁵	b⁶	ab⁶	b⁷	ab⁷
a	a	a²	ab	a³	ab⁶	ab²	ab	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵
b	b	b⁶	b²	b⁴	b⁷	b³	b²	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶
a²	a²	a³	ab⁶	ab	ab⁴	ab⁷	ab⁶	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³
ab	ab	ab⁶	ab²	ab⁴	ab⁷	ab³	ab²	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶
b²	b²	b⁷	b³	b⁵	b	b⁴	b³	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷
a³	a³	ab	ab⁴	ab⁶	ab²	ab⁵	ab⁴	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab
ab²	ab²	ab⁷	ab³	ab⁵	ab	ab⁴	ab³	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷
b³	b³	b	b⁴	b⁶	b²	b⁵	b⁴	b³	b⁶	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷	b³	b
ab³	ab³	ab	ab⁴	ab⁶	ab²	ab⁵	ab⁴	ab³	ab⁶	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab
b⁴	b⁴	b²	b⁵	b⁷	b³	b⁶	b⁵	b⁴	b⁷	b⁵	b	b⁶	b²	b⁷	b³	b	b⁴	b²
ab⁴	ab⁴	ab²	ab⁵	ab⁷	ab³	ab⁶	ab⁵	ab⁴	ab⁷	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²
b⁵	b⁵	b³	b⁶	b	b⁴	b⁷	b⁶	b⁵	b	b⁶	b²	b⁷	b³	b	b⁴	b²	b⁵	b³
ab⁵	ab⁵	ab³	ab⁶	ab	ab⁴	ab⁷	ab⁶	ab⁵	ab	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³
b⁶	b⁶	b⁴	b⁷	b²	b⁵	b	b⁷	b⁶	b²	b⁷	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴
ab⁶	ab⁶	ab⁴	ab⁷	ab²	ab⁵	ab	ab⁷	ab⁶	ab²	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴
b⁷	b⁷	b⁵	b	b³	b⁶	b²	b	b⁷	b³	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵
ab⁷	ab⁷	ab⁵	ab	ab³	ab⁶	ab²	ab	ab⁷	ab³	ab	ab⁴	ab²	ab⁵	ab³	ab⁶	ab⁴	ab⁷	ab⁵

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁸ ⇒ b	[10]
2.	ba ⇒ b⁶	[9]
3.	a²b ⇒ ab⁶	[11]
4.	a⁴ ⇒ ab	[1]

# ab:aaaa=ab,babb=b b/a
bbbbbbbb=b
ba=bbbbbb
aab=abbbbbb
aaaa=ab

Same cardinality

17 unique, 102 total

Σ	#	Presentation	Description	Related
8	1121	⟨a, b \| aa=1, abbba=b⟩	Finite non-commutative monoid with 18 elements	17 iso
9	3387	⟨a, b \| aa=1, abbbbb=b⟩	Finite non-commutative monoid with 18 elements	22 iso, 9 anti-iso
10	5521	⟨a, b \| aaab=1, bbbbbb=1⟩	Isomorphic to ℤ₁₈	33 iso
10	6732	⟨a, b \| aba=a, aaab=bb⟩	Finite non-commutative monoid with 18 elements
10	6788	⟨a, b \| aba=b, baab=aa⟩	Finite non-commutative monoid with 18 elements
10	6795	⟨a, b \| aba=b, bbbb=aa⟩	Finite non-commutative monoid with 18 elements
10	7039	⟨a, b \| bb=aa, aaaba=b⟩	Finite non-commutative monoid with 18 elements	2 iso
10	8910	⟨a, b \| aa=a, bbbbb=ab⟩	Finite non-commutative monoid with 18 elements	1 iso
11	15797	⟨a, b \| aab=bb, bbbba=a⟩	Finite non-commutative monoid with 18 elements
11	16042	⟨a, b \| aaa=ab, bbbb=ab⟩	Finite non-commutative monoid with 18 elements
11	16043	⟨a, b \| aaa=ab, bbbb=ba⟩	Finite non-commutative monoid with 18 elements
11	16313	⟨a, b \| aab=bb, baba=aa⟩	Finite non-commutative monoid with 18 elements
11	18758	⟨a, b \| aaa=a, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 6)}
11	18830	⟨a, b \| aaa=b, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 1)}
11	18831	⟨a, b \| aaa=b, bbbbbb=b⟩	Isomorphic to ℕ_{(18 = 3)}
11	19624	⟨a, b \| aab=b, bbbba=aa⟩	Finite non-commutative monoid with 18 elements
11	20914	⟨a, b \| bb=aa, ababa=aa⟩	Finite non-commutative monoid with 18 elements	1 iso

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	12191	⟨a, b \| aaaa=ab, bbab=b⟩	φ(a) = a, φ(b) = b
11	12193	⟨a, b \| aaaa=ab, bbba=b⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#12186 ⟨a, b \| aaaa=ab, babb=a⟩
Next:	#12188 ⟨a, b \| aaaa=ab, bbaa=a⟩

index 1, period 1	2 elements	ab², b⁷
index 1, period 7	12 elements	b, ab, b², b³, ab³, ...
index 2, period 7	2 elements	a², a³
index 4, period 7	1 element	a

#12187 ⟨a, b | aaaa=ab, babb=b⟩

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