#19638 ⟨a, b | aba=a, aaaab=bb⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 24 elements

Element profile

1 element center:
- 1
5 idempotent elements:
- 1, ab, ba, a⁵, a⁶b

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(a) = 0; deg(b) = 1

a⁹ ⇒ a⁴
ba⁴ ⇒ a⁸
aba ⇒ a
b² ⇒ a⁴b

# ab:aba=a,aaaab=bb a/b
aaaaaaaaa=aaaa
baaaa=aaaaaaaa
aba=a
bb=aaaab

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
1	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a	a	a²	ab	a³	a²b	a	a⁴	a³b	a²	ab	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a³b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
b	b	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a²	a²	a³	a²b	a⁴	a³b	a²	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ab	ab	a	a⁵b	a²	ab	a⁵	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
ba	ba	ba²	bab	ba³	ba²b	ba	a⁸	ba³b	ba²	bab	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a³	a³	a⁴	a³b	a⁵	a⁴b	a³	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a²b	a²b	a²	a⁶b	a³	a²b	a⁶	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
ba²	ba²	ba³	ba²b	a⁸	ba³b	ba²	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
bab	bab	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a³b	a³b	a³	a⁷b	a⁴	a³b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba³	ba³	a⁸	ba³b	a⁴	a⁸b	ba³	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba²b	ba²b	ba²	a⁵b	ba³	ba²b	a⁵	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁵	a⁵	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁴b	a⁴b	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
ba³b	ba³b	ba³	a⁶b	a⁸	ba³b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁶	a⁶	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁵b	a⁵b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁷	a⁷	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
a⁶b	a⁶b	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁸	a⁸	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a⁷b	a⁷b	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁸b	a⁸b	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

6 unique, 77 total

Σ	#	Presentation	Description	Related
8	425	⟨a, b \| aab=ba, bbb=1⟩	Finite non-commutative monoid with 24 elements	18 iso, 3 anti-iso
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Finite non-Abelian group with 24 elements	30 iso
10	6935	⟨a, b \| bb=aa, aaabab=1⟩	Finite non-Abelian group with 24 elements	2 iso
11	11515	⟨a, b \| ababa=b, abbaa=1⟩	Finite non-Abelian group with 24 elements	10 iso
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Isomorphic to ℤ₂₄	8 iso
11	20252	⟨a, b \| aba=b, aaaa=bab⟩	Finite non-commutative monoid with 24 elements

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