#8910 ⟨a, b | aa=a, bbbbb=ab⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 18 elements

Element profile

1 element center:
- 1
4 idempotent elements:
- 1, a, b⁸, b⁸a

Complete 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

b⁹ ⇒ b⁵
ab ⇒ b⁵
a² ⇒ a

# ab:aa=a,bbbbb=ab b/a
bbbbbbbbb=bbbbb
ab=bbbbb
aa=a

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
a	a	a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
ba	ba	ba	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b²a	b²a	b²a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b³a	b³a	b³a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁴a	b⁴a	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b⁵a	b⁵a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b⁶a	b⁶a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁷a	b⁷a	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a
b⁸	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁸a	b⁸a	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

17 unique, 103 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
11	12187	⟨a, b \| aaaa=ab, babb=b⟩	Finite non-commutative monoid with 18 elements	2 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

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	25255	⟨a, b \| aa=a, bbbbb=aab⟩	φ(a) = a, φ(b) = b

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