#3375 ⟨a, b | aa=1, ababba=b⟩

Properties

Presentation has sum-of-sides 9
Finite non-commutative monoid with 22 elements

Element profile

2 element center:
- 1, b⁴
2 idempotent elements:
- 1, b⁴

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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a² ⇒ 1
bab² ⇒ aba
b³a ⇒ (ab)³
b(ba)² ⇒ ab²ab
(ba)³ ⇒ ab³
b⁵ ⇒ b

# ab:aa=1,ababba=b reversed:a/b
aa=1
babb=aba
bbba=ababab
bbaba=abbab
bababa=abbb
bbbbb=b

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
1	1	a	b	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
a	a	1	ab	b	aba	ab²	ba	b²	(ab)²	ab²a	ab³	bab	b²a	b³	a(ba)²	ab²ab	ab⁴	(ba)²	b²ab	b⁴	(ab)³	b(ab)²
b	b	ba	b²	bab	b²a	b³	(ba)²	aba	b²ab	(ab)³	b⁴	b(ab)²	ab	(ab)²	ab²ab	ab²a	b	ab³	ab²	ba	a(ba)²	ab⁴
ab	ab	aba	ab²	(ab)²	ab²a	ab³	a(ba)²	ba	ab²ab	b(ab)²	ab⁴	(ab)³	b	bab	b²ab	b²a	ab	b³	b²	aba	(ba)²	b⁴
ba	ba	b	bab	b²	(ba)²	aba	b²a	b³	b(ab)²	ab	(ab)²	b²ab	(ab)³	b⁴	ab³	ab²	ba	ab²ab	ab²a	b	ab⁴	a(ba)²
b²	b²	b²a	b³	b²ab	(ab)³	b⁴	ab²ab	(ba)²	ab²a	ab⁴	b	a(ba)²	bab	b(ab)²	ab²	ab	b²	(ab)²	aba	b²a	ab³	ba
aba	aba	ab	(ab)²	ab²	a(ba)²	ba	ab²a	ab³	(ab)³	b	bab	ab²ab	b(ab)²	ab⁴	b³	b²	aba	b²ab	b²a	ab	b⁴	(ba)²
ab²	ab²	ab²a	ab³	ab²ab	b(ab)²	ab⁴	b²ab	a(ba)²	b²a	b⁴	ab	(ba)²	(ab)²	(ab)³	b²	b	ab²	bab	ba	ab²a	b³	aba
bab	bab	(ba)²	aba	b(ab)²	ab	(ab)²	ab³	b²a	ab²	a(ba)²	ba	ab⁴	b²	b²ab	ab²a	(ab)³	bab	b⁴	b³	(ba)²	ab²ab	b
b²a	b²a	b²	b²ab	b³	ab²ab	(ba)²	(ab)³	b⁴	a(ba)²	bab	b(ab)²	ab²a	ab⁴	b	(ab)²	aba	b²a	ab²	ab	b²	ba	ab³
b³	b³	(ab)³	b⁴	ab²a	ab⁴	b	ab²	ab²ab	ab	ba	b²	ab³	b²ab	a(ba)²	aba	bab	b³	b(ab)²	(ba)²	(ab)³	(ab)²	b²a
(ab)²	(ab)²	a(ba)²	ba	(ab)³	b	bab	b³	ab²a	b²	(ba)²	aba	b⁴	ab²	ab²ab	b²a	b(ab)²	(ab)²	ab⁴	ab³	a(ba)²	b²ab	ab
ab²a	ab²a	ab²	ab²ab	ab³	b²ab	a(ba)²	b(ab)²	ab⁴	(ba)²	(ab)²	(ab)³	b²a	b⁴	ab	bab	ba	ab²a	b²	b	ab²	aba	b³
ab³	ab³	b(ab)²	ab⁴	b²a	b⁴	ab	b²	b²ab	b	aba	ab²	b³	ab²ab	(ba)²	ba	(ab)²	ab³	(ab)³	a(ba)²	b(ab)²	bab	ab²a
(ba)²	(ba)²	bab	b(ab)²	aba	ab³	b²a	ab	(ab)²	ab⁴	b²	b²ab	ab²	a(ba)²	ba	b⁴	b³	(ba)²	ab²a	(ab)³	bab	b	ab²ab
b²ab	b²ab	ab²ab	(ba)²	a(ba)²	bab	b(ab)²	(ab)²	(ab)³	aba	ab³	b²a	ba	b³	ab²a	ab	ab⁴	b²ab	b	b⁴	ab²ab	ab²	b²
b⁴	b⁴	ab⁴	b	ab	ba	b²	aba	ab²	bab	b²a	b³	(ab)²	ab²a	ab³	(ba)²	b²ab	b⁴	a(ba)²	ab²ab	ab⁴	b(ab)²	(ab)³
a(ba)²	a(ba)²	(ab)²	(ab)³	ba	b³	ab²a	b	bab	b⁴	ab²	ab²ab	b²	(ba)²	aba	ab⁴	ab³	a(ba)²	b²a	b(ab)²	(ab)²	ab	b²ab
ab²ab	ab²ab	b²ab	a(ba)²	(ba)²	(ab)²	(ab)³	bab	b(ab)²	ba	b³	ab²a	aba	ab³	b²a	b	b⁴	ab²ab	ab	ab⁴	b²ab	b²	ab²
ab⁴	ab⁴	b⁴	ab	b	aba	ab²	ba	b²	(ab)²	ab²a	ab³	bab	b²a	b³	a(ba)²	ab²ab	ab⁴	(ba)²	b²ab	b⁴	(ab)³	b(ab)²
b(ab)²	b(ab)²	ab³	b²a	ab⁴	b²	b²ab	b⁴	ab	b³	ab²ab	(ba)²	b	aba	ab²	(ab)³	a(ba)²	b(ab)²	ba	(ab)²	ab³	ab²a	bab
(ab)³	(ab)³	b³	ab²a	b⁴	ab²	ab²ab	ab⁴	b	ab³	b²ab	a(ba)²	ab	ba	b²	b(ab)²	(ba)²	(ab)³	aba	bab	b³	b²a	(ab)²

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

3 unique, 8 total

Σ	#	Presentation	Description	Related
10	9759	⟨a, b \| aa=1, abbbbbb=b⟩	Finite non-commutative monoid with 22 elements	5 iso
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Finite non-commutative monoid with 22 elements
11	24722	⟨a, b \| aa=a, bbbbbb=ab⟩	Finite non-commutative monoid with 22 elements

Other isomorphic instances

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

15 total

Σ	#	Presentation	Mapping
9	3511	⟨a, b \| aa=1, ababb=ba⟩	φ(a) = a, φ(b) = b
9	3668	⟨a, b \| aa=1, babb=aba⟩	φ(a) = a, φ(b) = b
11	26483	⟨a, b \| aa=1, aaababba=b⟩	φ(a) = a, φ(b) = b
11	26565	⟨a, b \| aa=1, abaaabba=b⟩	φ(a) = a, φ(b) = b
11	27012	⟨a, b \| aa=1, aaababb=ba⟩	φ(a) = a, φ(b) = b
11	27050	⟨a, b \| aa=1, aababba=ab⟩	φ(a) = a, φ(b) = b
11	27093	⟨a, b \| aa=1, abaaabb=ba⟩	φ(a) = a, φ(b) = b
11	27109	⟨a, b \| aa=1, ababaab=ba⟩	φ(a) = a, φ(b) = b
11	27576	⟨a, b \| aa=1, aababb=aba⟩	φ(a) = a, φ(b) = b
11	27643	⟨a, b \| aa=1, ababba=aab⟩	φ(a) = a, φ(b) = b
11	27646	⟨a, b \| aa=1, ababba=baa⟩	φ(a) = a, φ(b) = b
11	27704	⟨a, b \| aa=1, baaabb=aba⟩	φ(a) = a, φ(b) = b
11	28061	⟨a, b \| aa=1, aaaba=babb⟩	φ(a) = a, φ(b) = b
11	28168	⟨a, b \| aa=1, ababb=aaba⟩	φ(a) = a, φ(b) = b
11	28174	⟨a, b \| aa=1, ababb=baaa⟩	φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

12 total

Σ	#	Presentation	Mapping
9	3515	⟨a, b \| aa=1, abbab=ba⟩	φ(a) = a, φ(b) = b
11	26491	⟨a, b \| aa=1, aaabbaba=b⟩	φ(a) = a, φ(b) = b
11	26571	⟨a, b \| aa=1, abaababa=b⟩	φ(a) = a, φ(b) = b
11	27020	⟨a, b \| aa=1, aaabbab=ba⟩	φ(a) = a, φ(b) = b
11	27062	⟨a, b \| aa=1, aabbaba=ab⟩	φ(a) = a, φ(b) = b
11	27097	⟨a, b \| aa=1, abaabab=ba⟩	φ(a) = a, φ(b) = b
11	27132	⟨a, b \| aa=1, abbaaab=ba⟩	φ(a) = a, φ(b) = b
11	27590	⟨a, b \| aa=1, aabbab=aba⟩	φ(a) = a, φ(b) = b
11	27712	⟨a, b \| aa=1, baabab=aba⟩	φ(a) = a, φ(b) = b
11	28063	⟨a, b \| aa=1, aaaba=bbab⟩	φ(a) = a, φ(b) = b
11	28184	⟨a, b \| aa=1, abbab=aaba⟩	φ(a) = a, φ(b) = b
11	28190	⟨a, b \| aa=1, abbab=baaa⟩	φ(a) = a, φ(b) = b

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