#7089 ⟨a, b | ab=aa, baaa=bb⟩

Properties

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

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, a⁴, ba³

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

a⁵ ⇒ a³
ab ⇒ a²
b² ⇒ ba³

# ab:ab=aa,baaa=bb a/b
aaaaa=aaa
ab=aa
bb=baaa

Cayley table

Idempotents are shown in bold.

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

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

31 unique, 916 total

Σ	#	Presentation	Description	Related
7	332	⟨a, b \| aa=1, abbb=b⟩	Finite non-commutative monoid with 10 elements	51 iso, 34 anti-iso
8	507	⟨a, b \| aaa=b, abbb=1⟩	Isomorphic to ℤ₁₀	536 iso
8	656	⟨a, b \| bb=aa, aba=a⟩	Finite non-commutative monoid with 10 elements	1 iso
8	657	⟨a, b \| bb=aa, aba=b⟩	Finite non-commutative monoid with 10 elements	1 iso
8	970	⟨a, b \| aa=a, bbb=ab⟩	Finite non-commutative monoid with 10 elements	3 iso
9	1654	⟨a, b \| aab=bb, bba=a⟩	Finite non-commutative monoid with 10 elements	6 iso
9	2012	⟨a, b \| aaa=b, abbb=a⟩	Isomorphic to ℕ_{(10 = 1)}	59 iso
9	2013	⟨a, b \| aaa=b, abbb=b⟩	Isomorphic to ℕ_{(10 = 3)}	35 iso
9	2277	⟨a, b \| bb=aa, aba=aa⟩	Finite non-commutative monoid with 10 elements	1 iso
9	2894	⟨a, b \| aa=a, bbbbb=a⟩	Isomorphic to ℕ_{(10 = 5)}	17 iso
9	2935	⟨a, b \| aa=b, bbbbb=b⟩	Isomorphic to ℕ_{(10 = 2)}	57 iso
10	4637	⟨a, b \| aaaa=b, aabb=b⟩	Isomorphic to ℕ_{(10 = 4)}	20 iso
10	5346	⟨a, b \| aaa=bb, abb=ab⟩	Finite non-commutative monoid with 10 elements	2 iso, 1 anti-iso
10	5356	⟨a, b \| aab=aa, aba=bb⟩	Finite non-commutative monoid with 10 elements
10	6587	⟨a, b \| aaa=b, abbb=bb⟩	Isomorphic to ℕ_{(10 = 6)}	3 iso
10	6633	⟨a, b \| aab=a, baaa=bb⟩	Finite non-commutative monoid with 10 elements	20 iso, 1 anti-iso
10	8619	⟨a, b \| aa=a, abbbbb=b⟩	Finite non-commutative monoid with 10 elements	5 iso
10	9051	⟨a, b \| ab=a, baaaa=bb⟩	Finite non-commutative monoid with 10 elements	4 iso
11	10729	⟨a, b \| aaab=baa, abab=1⟩	Finite non-Abelian group with 10 elements	3 iso
11	12157	⟨a, b \| aaaa=aa, abba=b⟩	Finite non-commutative monoid with 10 elements
11	12181	⟨a, b \| aaaa=ab, baaa=b⟩	Finite non-commutative monoid with 10 elements	1 iso
11	12196	⟨a, b \| aaaa=bb, aaab=a⟩	Finite commutative monoid with 10 elements	7 iso
11	12197	⟨a, b \| aaaa=bb, aaab=b⟩	Finite commutative monoid with 10 elements	1 iso
11	12452	⟨a, b \| aabb=aa, bbbb=a⟩	Isomorphic to ℕ_{(10 = 8)}	3 iso
11	15738	⟨a, b \| aab=bb, aaaaa=b⟩	Isomorphic to ℕ_{(10 = 7)}	7 iso
11	19564	⟨a, b \| aab=b, abbbb=aa⟩	Finite commutative monoid with 10 elements	5 iso
11	19899	⟨a, b \| aaa=b, abbb=bbb⟩	Isomorphic to ℕ_{(10 = 9)}	1 iso
11	20052	⟨a, b \| aab=b, aaaa=baa⟩	Finite non-commutative monoid with 10 elements
11	20253	⟨a, b \| aba=b, aaaa=bbb⟩	Finite commutative monoid with 10 elements
11	25905	⟨a, b \| ab=a, bbbb=baaa⟩	Finite non-commutative monoid with 10 elements
11	25909	⟨a, b \| ab=a, bbbb=bbaa⟩	Finite non-commutative monoid with 10 elements

Other isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
10	7093	⟨a, b \| ab=aa, baab=bb⟩	φ(a) = a, φ(b) = b
10	7097	⟨a, b \| ab=aa, baba=bb⟩	φ(a) = a, φ(b) = b
10	7101	⟨a, b \| ab=aa, babb=bb⟩	φ(a) = a, φ(b) = b

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