#425 ⟨a, b | aab=ba, bbb=1⟩

Properties

Presentation has sum-of-sides 8
Finite non-commutative monoid with 24 elements

Element profile

2 element center:
- 1, a⁷
2 idempotent elements:
- 1, a⁷

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

a⁸ ⇒ a
ab ⇒ ba⁴
b³ ⇒ 1

# ab:aab=ba,bbb=1 a/b
aaaaaaaa=a
ab=baaaa
bbb=1

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	a⁷	ba⁶	b²a⁵	ba⁷	b²a⁶	b²a⁷
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	a⁷	ba⁶	b²a⁵	ba⁷	b²a⁶	b²a⁷
a	a	a²	ba⁴	a³	ba⁵	b²a²	a⁴	ba⁶	b²a³	a⁵	ba⁷	b²a⁴	a⁶	ba	b²a⁵	a⁷	ba²	b²a⁶	a	ba³	b²a⁷	ba⁴	b²a	b²a²
b	b	ba	b²	ba²	b²a	1	ba³	b²a²	a	ba⁴	b²a³	a²	ba⁵	b²a⁴	a³	ba⁶	b²a⁵	a⁴	ba⁷	b²a⁶	a⁵	b²a⁷	a⁶	a⁷
a²	a²	a³	ba	a⁴	ba²	b²a⁴	a⁵	ba³	b²a⁵	a⁶	ba⁴	b²a⁶	a⁷	ba⁵	b²a⁷	a	ba⁶	b²a	a²	ba⁷	b²a²	ba	b²a³	b²a⁴
ba	ba	ba²	b²a⁴	ba³	b²a⁵	a²	ba⁴	b²a⁶	a³	ba⁵	b²a⁷	a⁴	ba⁶	b²a	a⁵	ba⁷	b²a²	a⁶	ba	b²a³	a⁷	b²a⁴	a	a²
b²	b²	b²a	1	b²a²	a	b	b²a³	a²	ba	b²a⁴	a³	ba²	b²a⁵	a⁴	ba³	b²a⁶	a⁵	ba⁴	b²a⁷	a⁶	ba⁵	a⁷	ba⁶	ba⁷
a³	a³	a⁴	ba⁵	a⁵	ba⁶	b²a⁶	a⁶	ba⁷	b²a⁷	a⁷	ba	b²a	a	ba²	b²a²	a²	ba³	b²a³	a³	ba⁴	b²a⁴	ba⁵	b²a⁵	b²a⁶
ba²	ba²	ba³	b²a	ba⁴	b²a²	a⁴	ba⁵	b²a³	a⁵	ba⁶	b²a⁴	a⁶	ba⁷	b²a⁵	a⁷	ba	b²a⁶	a	ba²	b²a⁷	a²	b²a	a³	a⁴
b²a	b²a	b²a²	a⁴	b²a³	a⁵	ba²	b²a⁴	a⁶	ba³	b²a⁵	a⁷	ba⁴	b²a⁶	a	ba⁵	b²a⁷	a²	ba⁶	b²a	a³	ba⁷	a⁴	ba	ba²
a⁴	a⁴	a⁵	ba²	a⁶	ba³	b²a	a⁷	ba⁴	b²a²	a	ba⁵	b²a³	a²	ba⁶	b²a⁴	a³	ba⁷	b²a⁵	a⁴	ba	b²a⁶	ba²	b²a⁷	b²a
ba³	ba³	ba⁴	b²a⁵	ba⁵	b²a⁶	a⁶	ba⁶	b²a⁷	a⁷	ba⁷	b²a	a	ba	b²a²	a²	ba²	b²a³	a³	ba³	b²a⁴	a⁴	b²a⁵	a⁵	a⁶
b²a²	b²a²	b²a³	a	b²a⁴	a²	ba⁴	b²a⁵	a³	ba⁵	b²a⁶	a⁴	ba⁶	b²a⁷	a⁵	ba⁷	b²a	a⁶	ba	b²a²	a⁷	ba²	a	ba³	ba⁴
a⁵	a⁵	a⁶	ba⁶	a⁷	ba⁷	b²a³	a	ba	b²a⁴	a²	ba²	b²a⁵	a³	ba³	b²a⁶	a⁴	ba⁴	b²a⁷	a⁵	ba⁵	b²a	ba⁶	b²a²	b²a³
ba⁴	ba⁴	ba⁵	b²a²	ba⁶	b²a³	a	ba⁷	b²a⁴	a²	ba	b²a⁵	a³	ba²	b²a⁶	a⁴	ba³	b²a⁷	a⁵	ba⁴	b²a	a⁶	b²a²	a⁷	a
b²a³	b²a³	b²a⁴	a⁵	b²a⁵	a⁶	ba⁶	b²a⁶	a⁷	ba⁷	b²a⁷	a	ba	b²a	a²	ba²	b²a²	a³	ba³	b²a³	a⁴	ba⁴	a⁵	ba⁵	ba⁶
a⁶	a⁶	a⁷	ba³	a	ba⁴	b²a⁵	a²	ba⁵	b²a⁶	a³	ba⁶	b²a⁷	a⁴	ba⁷	b²a	a⁵	ba	b²a²	a⁶	ba²	b²a³	ba³	b²a⁴	b²a⁵
ba⁵	ba⁵	ba⁶	b²a⁶	ba⁷	b²a⁷	a³	ba	b²a	a⁴	ba²	b²a²	a⁵	ba³	b²a³	a⁶	ba⁴	b²a⁴	a⁷	ba⁵	b²a⁵	a	b²a⁶	a²	a³
b²a⁴	b²a⁴	b²a⁵	a²	b²a⁶	a³	ba	b²a⁷	a⁴	ba²	b²a	a⁵	ba³	b²a²	a⁶	ba⁴	b²a³	a⁷	ba⁵	b²a⁴	a	ba⁶	a²	ba⁷	ba
a⁷	a⁷	a	ba⁷	a²	ba	b²a⁷	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	a⁷	ba⁶	b²a⁵	ba⁷	b²a⁶	b²a⁷
ba⁶	ba⁶	ba⁷	b²a³	ba	b²a⁴	a⁵	ba²	b²a⁵	a⁶	ba³	b²a⁶	a⁷	ba⁴	b²a⁷	a	ba⁵	b²a	a²	ba⁶	b²a²	a³	b²a³	a⁴	a⁵
b²a⁵	b²a⁵	b²a⁶	a⁶	b²a⁷	a⁷	ba³	b²a	a	ba⁴	b²a²	a²	ba⁵	b²a³	a³	ba⁶	b²a⁴	a⁴	ba⁷	b²a⁵	a⁵	ba	a⁶	ba²	ba³
ba⁷	ba⁷	ba	b²a⁷	ba²	b²a	a⁷	ba³	b²a²	a	ba⁴	b²a³	a²	ba⁵	b²a⁴	a³	ba⁶	b²a⁵	a⁴	ba⁷	b²a⁶	a⁵	b²a⁷	a⁶	a⁷
b²a⁶	b²a⁶	b²a⁷	a³	b²a	a⁴	ba⁵	b²a²	a⁵	ba⁶	b²a³	a⁶	ba⁷	b²a⁴	a⁷	ba	b²a⁵	a	ba²	b²a⁶	a²	ba³	a³	ba⁴	ba⁵
b²a⁷	b²a⁷	b²a	a⁷	b²a²	a	ba⁷	b²a³	a²	ba	b²a⁴	a³	ba²	b²a⁵	a⁴	ba³	b²a⁶	a⁵	ba⁴	b²a⁷	a⁶	ba⁵	a⁷	ba⁶	ba⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

6 unique, 56 total

Σ	#	Presentation	Description	Related
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Finite non-Abelian group with 24 elements	30 iso
10	6935	⟨a, b \| bb=aa, aaabab=1⟩	Finite non-Abelian group with 24 elements	2 iso
11	11515	⟨a, b \| ababa=b, abbaa=1⟩	Finite non-Abelian group with 24 elements	10 iso
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Isomorphic to ℤ₂₄	8 iso
11	19638	⟨a, b \| aba=a, aaaab=bb⟩	Finite non-commutative monoid with 24 elements
11	20252	⟨a, b \| aba=b, aaaa=bab⟩	Finite non-commutative monoid with 24 elements

Other isomorphic instances

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

18 total

Σ	#	Presentation	Mapping
9	2429	⟨a, b \| aaa=1, aabba=b⟩	φ(a) = b, φ(b) = a
9	2570	⟨a, b \| aaa=1, aaba=bb⟩	φ(a) = bb, φ(b) = a
9	2577	⟨a, b \| aaa=1, abab=ba⟩	φ(a) = bb, φ(b) = ba
10	7784	⟨a, b \| aaa=1, abaab=ba⟩	φ(a) = bb, φ(b) = bba
10	8043	⟨a, b \| aaa=1, aaba=bab⟩	φ(a) = bb, φ(b) = ba
10	8048	⟨a, b \| aaa=1, aabb=aba⟩	φ(a) = bb, φ(b) = a
10	8050	⟨a, b \| aaa=1, aabb=baa⟩	φ(a) = b, φ(b) = a
10	8063	⟨a, b \| aaa=1, abba=aab⟩	φ(a) = b, φ(b) = a
11	21683	⟨a, b \| aaa=1, aabaaba=b⟩	φ(a) = b, φ(b) = ba
11	22205	⟨a, b \| aaa=1, aaaabb=ba⟩	φ(a) = bb, φ(b) = a
11	22216	⟨a, b \| aaa=1, aaabba=ab⟩	φ(a) = b, φ(b) = a
11	22236	⟨a, b \| aaa=1, aabbaa=ab⟩	φ(a) = bb, φ(b) = a
11	22252	⟨a, b \| aaa=1, abaaab=ba⟩	φ(a) = bb, φ(b) = a
11	22732	⟨a, b \| aaa=1, aaaab=bba⟩	φ(a) = b, φ(b) = a
11	22737	⟨a, b \| aaa=1, aaaba=abb⟩	φ(a) = bb, φ(b) = a
11	22753	⟨a, b \| aaa=1, aabaa=abb⟩	φ(a) = b, φ(b) = a
11	22758	⟨a, b \| aaa=1, aabab=aba⟩	φ(a) = bb, φ(b) = ba
11	23296	⟨a, b \| aaa=1, baab=aaba⟩	φ(a) = bb, φ(b) = bba

Other anti-isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
10	7499	⟨a, b \| aaa=1, aababa=b⟩	φ(a) = bb, φ(b) = ba
11	22760	⟨a, b \| aaa=1, aabab=baa⟩	φ(a) = bb, φ(b) = ba
11	22789	⟨a, b \| aaa=1, ababa=aab⟩	φ(a) = bb, φ(b) = ba

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