#9375 ⟨a, b | ab=a, bbba=bbb⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 7 elements

Element profile

1 element center:
- 1
5 idempotent elements:
- 1, a, ba, b²a, b³
4 left zero elements:
- a, ba, b²a, b³

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

a² ⇒ a
ab ⇒ a
b³a ⇒ b³
b⁴ ⇒ b³

# ab:ab=a,bbba=bbb ab
aa=a
ab=a
bbba=bbb
bbbb=bbb

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³
1	1	a	b	ba	b²	b²a	b³
a	a	a	a	a	a	a	a
b	b	ba	b²	b²a	b³	b³	b³
ba	ba	ba	ba	ba	ba	ba	ba
b²	b²	b²a	b³	b³	b³	b³	b³
b²a	b²a	b²a	b²a	b²a	b²a	b²a	b²a
b³	b³	b³	b³	b³	b³	b³	b³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

22 unique, 1220 total

Σ	#	Presentation	Description	Related
7	116	⟨a, b \| aaa=b, abb=1⟩	Isomorphic to ℤ₇	763 iso
8	577	⟨a, b \| aaa=b, abb=a⟩	Isomorphic to ℕ_{(7 = 1)}	59 iso
8	578	⟨a, b \| aaa=b, abb=b⟩	Isomorphic to ℕ_{(7 = 3)}	44 iso
8	913	⟨a, b \| aa=b, abbb=b⟩	Isomorphic to ℕ_{(7 = 2)}	82 iso
8	937	⟨a, b \| ab=a, baaa=b⟩	Finite non-commutative monoid with 7 elements	54 iso, 12 anti-iso
9	1593	⟨a, b \| aaa=ab, baa=b⟩	Finite non-commutative monoid with 7 elements	1 iso, 2 anti-iso
9	1601	⟨a, b \| aaa=bb, aab=b⟩	Finite commutative monoid with 7 elements	5 iso
9	1620	⟨a, b \| aab=aa, bbb=a⟩	Isomorphic to ℕ_{(7 = 6)}	35 iso
9	1632	⟨a, b \| aab=ab, bbb=a⟩	Isomorphic to ℕ_{(7 = 4)}	50 iso
9	2074	⟨a, b \| aab=b, babb=a⟩	Finite commutative monoid with 7 elements	20 iso
9	2231	⟨a, b \| ab=aa, aaa=bb⟩	Finite non-commutative monoid with 7 elements	3 iso
9	2271	⟨a, b \| bb=aa, aaa=ab⟩	Finite non-commutative monoid with 7 elements	1 iso, 2 anti-iso
9	3197	⟨a, b \| ab=a, bbb=baa⟩	Finite non-commutative monoid with 7 elements	5 iso
10	4162	⟨a, b \| abb=aab, bbb=a⟩	Isomorphic to ℕ_{(7 = 5)}	42 iso
10	6251	⟨a, b \| aaa=a, aabba=b⟩	Finite non-commutative monoid with 7 elements	1 iso
10	8987	⟨a, b \| ab=a, aaaaa=bb⟩	Finite commutative monoid with 7 elements	5 iso
10	9108	⟨a, b \| ab=a, bbbbb=aa⟩	Finite commutative monoid with 7 elements	2 iso
10	9110	⟨a, b \| ab=a, bbbbb=ba⟩	Finite non-commutative monoid with 7 elements	1 iso
10	9263	⟨a, b \| ab=a, aaaa=bbb⟩	Finite commutative monoid with 7 elements	4 iso
10	9376	⟨a, b \| ab=a, bbbb=aaa⟩	Finite commutative monoid with 7 elements	3 iso
10	9382	⟨a, b \| ab=a, bbbb=bba⟩	Finite non-commutative monoid with 7 elements	1 iso
11	14843	⟨a, b \| abba=b, aabab=a⟩	Finite non-commutative monoid with 7 elements	1 iso

Other isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
11	20039	⟨a, b \| aab=a, bbba=bbb⟩	φ(a) = a, φ(b) = b
11	25643	⟨a, b \| ab=a, bbbab=bbb⟩	φ(a) = a, φ(b) = b
11	25651	⟨a, b \| ab=a, bbbba=bbb⟩	φ(a) = a, φ(b) = b

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Next:	#9376 ⟨a, b \| ab=a, bbbb=aaa⟩