#21023 ⟨a, b | ab=aa, bbaa=bbb⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 14 elements

Element profile

1 element center:
- 1
4 idempotent elements:
- 1, a⁴, b³a, ba⁴
3 left zero elements:
- a⁴, b³a, ba⁴

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
b²a² ⇒ b³
b⁴ ⇒ b³a
a⁵ ⇒ a⁴

# ab:ab=aa,bbaa=bbb ab
ab=aa
bbaa=bbb
bbbb=bbba
aaaaa=aaaa

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b³a	ba⁴
1	1	a	b	a²	ba	b²	a³	ba²	b²a	b³	a⁴	ba³	b³a	ba⁴
a	a	a²	a²	a³	a³	a³	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
b	b	ba	b²	ba²	b²a	b³	ba³	b³	b³a	b³a	ba⁴	b³a	b³a	b³a
a²	a²	a³	a³	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
ba	ba	ba²	ba²	ba³	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴
b²	b²	b²a	b³	b³	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a
a³	a³	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
ba²	ba²	ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴
b²a	b²a	b³	b³	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a
b³	b³	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a
a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴	a⁴
ba³	ba³	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴
b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a	b³a
ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴	ba⁴

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

30 unique, 315 total

Σ	#	Presentation	Description	Related
8	1123	⟨a, b \| aa=1, abbbb=b⟩	Finite non-commutative monoid with 14 elements	42 iso, 17 anti-iso
9	1682	⟨a, b \| aba=bb, bab=a⟩	Finite non-commutative monoid with 14 elements	1 iso
9	3034	⟨a, b \| aa=a, bbbb=ab⟩	Finite non-commutative monoid with 14 elements	2 iso
10	3773	⟨a, b \| aaaa=bb, abbb=1⟩	Isomorphic to ℤ₁₄	165 iso
10	5218	⟨a, b \| aab=bb, bbba=a⟩	Finite non-commutative monoid with 14 elements	5 iso, 1 anti-iso
10	5404	⟨a, b \| aab=bb, aba=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
10	6718	⟨a, b \| aab=b, bbba=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	12183	⟨a, b \| aaaa=ab, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12206	⟨a, b \| aaaa=bb, abbb=a⟩	Finite commutative monoid with 14 elements	1 iso
11	12207	⟨a, b \| aaaa=bb, abbb=b⟩	Finite commutative monoid with 14 elements	1 iso
11	12441	⟨a, b \| aabb=aa, baab=b⟩	Finite non-commutative monoid with 14 elements	3 iso
11	12499	⟨a, b \| abab=aa, abba=b⟩	Finite non-commutative monoid with 14 elements
11	14383	⟨a, b \| aaaa=b, aabbb=a⟩	Isomorphic to ℕ_{(14 = 1)}	15 iso
11	14384	⟨a, b \| aaaa=b, aabbb=b⟩	Isomorphic to ℕ_{(14 = 4)}	5 iso
11	15532	⟨a, b \| aaa=bb, abbbb=b⟩	Isomorphic to ℕ_{(14 = 3)}	11 iso
11	15539	⟨a, b \| aaa=bb, babbb=a⟩	Finite commutative monoid with 14 elements
11	16020	⟨a, b \| aaa=ab, baab=bb⟩	Finite non-commutative monoid with 14 elements	1 iso
11	16079	⟨a, b \| aaa=bb, bbbb=ab⟩	Finite non-commutative monoid with 14 elements
11	16293	⟨a, b \| aab=bb, abab=aa⟩	Finite non-commutative monoid with 14 elements
11	16470	⟨a, b \| aba=bb, bbbb=aa⟩	Finite non-commutative monoid with 14 elements
11	19552	⟨a, b \| aab=b, abbaa=aa⟩	Finite non-commutative monoid with 14 elements	2 iso
11	20844	⟨a, b \| ab=aa, bbbbb=aa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	20846	⟨a, b \| ab=aa, bbbbb=ba⟩	Finite non-commutative monoid with 14 elements
11	21040	⟨a, b \| ab=aa, bbbb=aaa⟩	Finite non-commutative monoid with 14 elements	3 iso
11	21044	⟨a, b \| ab=aa, bbbb=baa⟩	Finite non-commutative monoid with 14 elements	1 iso
11	21046	⟨a, b \| ab=aa, bbbb=bba⟩	Finite non-commutative monoid with 14 elements
11	21110	⟨a, b \| bb=aa, abab=aaa⟩	Finite non-commutative monoid with 14 elements	2 anti-iso
11	24186	⟨a, b \| aa=a, bbbbbbb=a⟩	Isomorphic to ℕ_{(14 = 7)}
11	24331	⟨a, b \| aa=b, bbbbbbb=b⟩	Isomorphic to ℕ_{(14 = 2)}
11	25055	⟨a, b \| ab=a, bbaaaa=bb⟩	Finite non-commutative monoid with 14 elements

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	21031	⟨a, b \| ab=aa, bbab=bbb⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#21022 ⟨a, b \| ab=aa, bbaa=bba⟩
Next:	#21032 ⟨a, b \| ab=aa, bbba=aaa⟩