#20787 ⟨a, b | ab=aa, baaaa=bb⟩

Properties

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

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, a³, ba⁵

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 ⇒ a³
ab ⇒ a²
b² ⇒ ba⁴

# ab:ab=aa,baaaa=bb a/b
aaaaaa=aaa
ab=aa
bb=baaaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	ba⁵
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	ba⁵
a	a	a²	a²	a³	a³	a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴
b	b	ba	ba⁴	ba²	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³
a²	a²	a³	a³	a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵
ba	ba	ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴
a³	a³	a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁵	a³
ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵
a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴
ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³
a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵	a⁵	a³	a³	a⁴	a⁴	a⁵
ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴
ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵	ba⁵	ba³	ba³	ba⁴	ba⁴	ba⁵

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

37 unique, 652 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	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	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.

7 total

Σ	#	Presentation	Mapping
11	20791	⟨a, b \| ab=aa, baaab=bb⟩	φ(a) = a, φ(b) = b
11	20795	⟨a, b \| ab=aa, baaba=bb⟩	φ(a) = a, φ(b) = b
11	20799	⟨a, b \| ab=aa, baabb=bb⟩	φ(a) = a, φ(b) = b
11	20803	⟨a, b \| ab=aa, babaa=bb⟩	φ(a) = a, φ(b) = b
11	20807	⟨a, b \| ab=aa, babab=bb⟩	φ(a) = a, φ(b) = b
11	20811	⟨a, b \| ab=aa, babba=bb⟩	φ(a) = a, φ(b) = b
11	20815	⟨a, b \| ab=aa, babbb=bb⟩	φ(a) = a, φ(b) = b

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