#13259 ⟨a, b | bab=aba, bbb=aa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 16 elements

Element profile

9 element center:
- 1, bab, b³, ab²a, ab³, b⁴, b⁵, b⁶, b⁷
2 idempotent elements:
- 1, b⁷
Zero element: b⁷

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

b⁸ ⇒ b⁷
ab⁴ ⇒ b⁷
bab² ⇒ b⁷
b²ab ⇒ b⁷
b³a ⇒ ab³
a² ⇒ b³
aba ⇒ bab

# ab:bab=aba,bbb=aa b/a
bbbbbbbb=bbbbbbb
abbbb=bbbbbbb
babb=bbbbbbb
bbab=bbbbbbb
bbba=abbb
aa=bbb
aba=bab

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab²a	ab³	b⁴	b⁵	b⁶	b⁷
1	1	a	b	ab	ba	b²	ab²	bab	b²a	b³	ab²a	ab³	b⁴	b⁵	b⁶	b⁷
a	a	b³	ab	b⁴	bab	ab²	b⁵	b⁷	ab²a	ab³	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷
b	b	ba	b²	bab	b²a	b³	b⁷	b⁷	ab³	b⁴	b⁷	b⁷	b⁵	b⁶	b⁷	b⁷
ab	ab	bab	ab²	b⁷	ab²a	ab³	b⁷	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
ba	ba	b⁴	bab	b⁵	b⁷	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b²	b²	b²a	b³	b⁷	ab³	b⁴	b⁷	b⁷	b⁷	b⁵	b⁷	b⁷	b⁶	b⁷	b⁷	b⁷
ab²	ab²	ab²a	ab³	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
bab	bab	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b²a	b²a	b⁵	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b³	b³	ab³	b⁴	b⁷	b⁷	b⁵	b⁷	b⁷	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
ab²a	ab²a	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
ab³	ab³	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b⁴	b⁴	b⁷	b⁵	b⁷	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b⁵	b⁵	b⁷	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b⁶	b⁶	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷
b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷	b⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

20 unique, 175 total

Σ	#	Presentation	Description	Related
8	628	⟨a, b \| bb=aa, abab=1⟩	Finite non-Abelian group with 16 elements	58 iso
9	1331	⟨a, b \| aaaa=b, bbbb=1⟩	Isomorphic to ℤ₁₆	67 iso
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Finite non-commutative monoid with 16 elements	4 anti-iso
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Finite non-Abelian group with 16 elements	7 iso
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Isomorphic to ℕ_{(16 = 4)}	1 iso
10	4648	⟨a, b \| aaaa=b, bbbb=a⟩	Isomorphic to ℕ_{(16 = 1)}	5 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(16 = 5)}	2 iso
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Finite non-commutative monoid with 16 elements	6 iso
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Finite non-commutative monoid with 16 elements
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Finite non-commutative monoid with 16 elements
11	16060	⟨a, b \| aaa=bb, abab=aa⟩	Finite non-commutative monoid with 16 elements
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Finite non-commutative monoid with 16 elements
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Isomorphic to ℕ_{(16 = 3)}	2 iso
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Finite non-commutative monoid with 16 elements
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Finite non-commutative monoid with 16 elements
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Finite non-commutative monoid with 16 elements

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