#645 ⟨a, b | ab=aa, bbb=b⟩

Properties

Presentation has sum-of-sides 8
Finite non-commutative monoid with 12 elements

Element profile

1 element center:
- 1
5 idempotent elements:
- 1, a², b², ba³, b²a²

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 shortlex with a < b

ab ⇒ a²
b³ ⇒ b
a⁴ ⇒ a²

# ab:ab=aa,bbb=b ab
ab=aa
bbb=b
aaaa=aa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	ba³	b²a²	b²a³
1	1	a	b	a²	ba	b²	a³	ba²	b²a	ba³	b²a²	b²a³
a	a	a²	a²	a³	a³	a³	a²	a²	a²	a³	a³	a²
b	b	ba	b²	ba²	b²a	b	ba³	b²a²	ba	b²a³	ba²	ba³
a²	a²	a³	a³	a²	a²	a²	a³	a³	a³	a²	a²	a³
ba	ba	ba²	ba²	ba³	ba³	ba³	ba²	ba²	ba²	ba³	ba³	ba²
b²	b²	b²a	b	b²a²	ba	b²	b²a³	ba²	b²a	ba³	b²a²	b²a³
a³	a³	a²	a²	a³	a³	a³	a²	a²	a²	a³	a³	a²
ba²	ba²	ba³	ba³	ba²	ba²	ba²	ba³	ba³	ba³	ba²	ba²	ba³
b²a	b²a	b²a²	b²a²	b²a³	b²a³	b²a³	b²a²	b²a²	b²a²	b²a³	b²a³	b²a²
ba³	ba³	ba²	ba²	ba³	ba³	ba³	ba²	ba²	ba²	ba³	ba³	ba²
b²a²	b²a²	b²a³	b²a³	b²a²	b²a²	b²a²	b²a³	b²a³	b²a³	b²a²	b²a²	b²a³
b²a³	b²a³	b²a²	b²a²	b²a³	b²a³	b²a³	b²a²	b²a²	b²a²	b²a³	b²a³	b²a²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 658 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
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	16448	⟨a, b \| aba=bb, aabb=aa⟩	Finite non-commutative monoid with 12 elements	2 iso
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 anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
10	6532	⟨a, b \| aaa=a, aaab=bb⟩	φ(a) = b, φ(b) = a

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