#16448 ⟨a, b | aba=bb, aabb=aa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 12 elements

Element profile

8 element center:
- 1, a², a³, aba, bab, a⁴, (ba)², a⁵
2 idempotent elements:
- 1, a⁴

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

a⁶ ⇒ a²
ba² ⇒ a⁴
a²b ⇒ a⁴
b² ⇒ aba
(ab)² ⇒ (ba)²

# ab:aba=bb,aabb=aa a/b
aaaaaa=aa
baa=aaaa
aab=aaaa
bb=aba
abab=baba

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	aba	bab	a⁴	(ba)²	a⁵
1	1	a	b	a²	ab	ba	a³	aba	bab	a⁴	(ba)²	a⁵
a	a	a²	ab	a³	a⁴	aba	a⁴	a⁵	(ba)²	a⁵	a³	a²
b	b	ba	aba	a⁴	bab	a⁵	a⁵	(ba)²	a³	a²	a⁴	a³
a²	a²	a³	a⁴	a⁴	a⁵	a⁵	a⁵	a²	a³	a²	a⁴	a³
ab	ab	aba	a⁵	a⁵	(ba)²	a²	a²	a³	a⁴	a³	a⁵	a⁴
ba	ba	a⁴	bab	a⁵	a²	(ba)²	a²	a³	a⁴	a³	a⁵	a⁴
a³	a³	a⁴	a⁵	a⁵	a²	a²	a²	a³	a⁴	a³	a⁵	a⁴
aba	aba	a⁵	(ba)²	a²	a³	a³	a³	a⁴	a⁵	a⁴	a²	a⁵
bab	bab	(ba)²	a³	a³	a⁴	a⁴	a⁴	a⁵	a²	a⁵	a³	a²
a⁴	a⁴	a⁵	a²	a²	a³	a³	a³	a⁴	a⁵	a⁴	a²	a⁵
(ba)²	(ba)²	a³	a⁴	a⁴	a⁵	a⁵	a⁵	a²	a³	a²	a⁴	a³
a⁵	a⁵	a²	a³	a³	a⁴	a⁴	a⁴	a⁵	a²	a⁵	a³	a²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 657 total

Σ	#	Presentation	Description	Related
7	124	⟨a, b \| aab=a, bbb=1⟩	Finite non-commutative monoid with 12 elements	23 iso, 38 anti-iso
8	458	⟨a, b \| aaaa=1, abbb=1⟩	Isomorphic to ℤ₁₂	325 iso
8	645	⟨a, b \| ab=aa, bbb=b⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
9	1646	⟨a, b \| aab=bb, aba=a⟩	Finite non-commutative monoid with 12 elements	2 iso
9	1963	⟨a, b \| aba=b, aaabb=1⟩	Finite non-Abelian group with 12 elements	30 iso
9	1998	⟨a, b \| aaa=a, bbbb=a⟩	Isomorphic to ℕ_{(12 = 4)}	6 iso
9	2018	⟨a, b \| aaa=b, bbbb=a⟩	Isomorphic to ℕ_{(12 = 1)}	27 iso
9	2019	⟨a, b \| aaa=b, bbbb=b⟩	Isomorphic to ℕ_{(12 = 3)}	23 iso
9	2259	⟨a, b \| ab=aa, bbb=bb⟩	Finite non-commutative monoid with 12 elements
10	5040	⟨a, b \| aaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 8)}
10	5072	⟨a, b \| aaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(12 = 5)}	7 iso
10	5092	⟨a, b \| aaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(12 = 2)}	43 iso
10	5202	⟨a, b \| aab=bb, abba=a⟩	Finite non-commutative monoid with 12 elements	9 iso, 32 anti-iso
10	5330	⟨a, b \| aaa=ab, bab=bb⟩	Finite non-commutative monoid with 12 elements	1 iso
10	6597	⟨a, b \| aaa=b, bbbb=bb⟩	Isomorphic to ℕ_{(12 = 6)}	4 iso
10	6660	⟨a, b \| aab=a, bbbb=ba⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	6661	⟨a, b \| aab=a, bbbb=bb⟩	Finite non-commutative monoid with 12 elements	1 anti-iso
10	7105	⟨a, b \| ab=aa, bbaa=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	12897	⟨a, b \| aab=aaa, baaa=b⟩	Finite non-commutative monoid with 12 elements	5 iso
11	12910	⟨a, b \| aab=aaa, bbbb=a⟩	Isomorphic to ℕ_{(12 = 9)}	2 iso
11	15428	⟨a, b \| aaa=aa, abbbb=b⟩	Finite non-commutative monoid with 12 elements
11	15996	⟨a, b \| aaa=ab, aabb=bb⟩	Finite non-commutative monoid with 12 elements	2 iso
11	16057	⟨a, b \| aaa=bb, aabb=ab⟩	Finite non-commutative monoid with 12 elements	2 iso, 2 anti-iso
11	16104	⟨a, b \| aab=aa, abab=bb⟩	Finite non-commutative monoid with 12 elements
11	16274	⟨a, b \| aab=bb, aaaa=ab⟩	Finite non-commutative monoid with 12 elements	1 iso
11	16275	⟨a, b \| aab=bb, aaaa=ba⟩	Finite non-commutative monoid with 12 elements
11	19773	⟨a, b \| aba=b, bbbbb=aa⟩	Finite commutative monoid with 12 elements
11	19917	⟨a, b \| aaa=b, bbbb=abb⟩	Isomorphic to ℕ_{(12 = 7)}	2 iso
11	20054	⟨a, b \| aab=b, aaaa=bba⟩	Finite non-commutative monoid with 12 elements
11	20686	⟨a, b \| bb=aa, aaaaab=a⟩	Finite commutative monoid with 12 elements	15 iso
11	20787	⟨a, b \| ab=aa, baaaa=bb⟩	Finite non-commutative monoid with 12 elements	7 iso
11	21086	⟨a, b \| bb=aa, aaaa=aba⟩	Finite non-commutative monoid with 12 elements	4 iso
11	21092	⟨a, b \| bb=aa, aaab=aba⟩	Finite non-commutative monoid with 12 elements	3 iso
11	21112	⟨a, b \| bb=aa, abab=aba⟩	Finite non-commutative monoid with 12 elements
11	24147	⟨a, b \| aa=a, abbbbbb=b⟩	Finite non-commutative monoid with 12 elements
11	24991	⟨a, b \| ab=a, baaaaa=bb⟩	Finite non-commutative monoid with 12 elements
11	25603	⟨a, b \| ab=a, bbaaa=bbb⟩	Finite non-commutative monoid with 12 elements

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	16456	⟨a, b \| aba=bb, abba=aa⟩	φ(a) = a, φ(b) = b
11	16463	⟨a, b \| aba=bb, baab=aa⟩	φ(a) = a, φ(b) = b

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