#6660 ⟨a, b | aab=a, bbbb=ba⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 12 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, ab, b⁴

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 ⇒ b⁴
ab⁴ ⇒ a
ba ⇒ b⁴
a² ⇒ ab³

# ab:aab=a,bbbb=ba b/a
bbbbbbbb=bbbb
abbbb=a
ba=bbbb
aa=abbb

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	b²	ab²	b³	ab³	b⁴	b⁵	b⁶	b⁷
1	1	a	b	ab	b²	ab²	b³	ab³	b⁴	b⁵	b⁶	b⁷
a	a	ab³	ab	a	ab²	ab	ab³	ab²	a	ab	ab²	ab³
b	b	b⁴	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b⁶	b⁷	b⁴
ab	ab	a	ab²	ab	ab³	ab²	a	ab³	ab	ab²	ab³	a
b²	b²	b⁵	b³	b⁶	b⁴	b⁷	b⁵	b⁴	b⁶	b⁷	b⁴	b⁵
ab²	ab²	ab	ab³	ab²	a	ab³	ab	a	ab²	ab³	a	ab
b³	b³	b⁶	b⁴	b⁷	b⁵	b⁴	b⁶	b⁵	b⁷	b⁴	b⁵	b⁶
ab³	ab³	ab²	a	ab³	ab	a	ab²	ab	ab³	a	ab	ab²
b⁴	b⁴	b⁷	b⁵	b⁴	b⁶	b⁵	b⁷	b⁶	b⁴	b⁵	b⁶	b⁷
b⁵	b⁵	b⁴	b⁶	b⁵	b⁷	b⁶	b⁴	b⁷	b⁵	b⁶	b⁷	b⁴
b⁶	b⁶	b⁵	b⁷	b⁶	b⁴	b⁷	b⁵	b⁴	b⁶	b⁷	b⁴	b⁵
b⁷	b⁷	b⁶	b⁴	b⁷	b⁵	b⁴	b⁶	b⁵	b⁷	b⁴	b⁵	b⁶

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
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	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	6760	⟨a, b \| aba=a, bbbb=ab⟩	φ(a) = a, φ(b) = b

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