⟨a, b | ab=a, baa=b⟩

Up:

	1	a	b	a²	ba
1	1	a	b	a²	ba
a	a	a²	a	a	a²
b	b	ba	b	b	ba
a²	a²	a	a²	a²	a
ba	ba	b	ba	ba	b

a²

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right shortlex with a < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ab ⇒ a	[1]
2.	b² ⇒ b	[4]
3.	a³ ⇒ a	[3]
4.	ba² ⇒ b	[2]

# ab:ab=a,baa=b ab - - ab=a bb=b aaa=a baa=b

Same cardinality

Σ	#	Presentation	Properties	Description	φ
6	44	⟨a, b \| aa=b, abb=1⟩	Grp Com 5	ℤ₅	1132
7	253	⟨a, b \| aa=b, abb=a⟩	Noc Com 5	ℕ_{(5 = 1)}	71
7	254	⟨a, b \| aa=b, abb=b⟩	Noc Com 5	ℕ_{(5 = 2)}	43
8	574	⟨a, b \| aaa=b, aab=b⟩	Noc Com 5	ℕ_{(5 = 3)}	27
8	950	⟨a, b \| ab=a, bbbb=a⟩	Noc Com 5	ℕ_{(5 = 4)}	32
8	995	⟨a, b \| ab=a, aaa=bb⟩	Noc Com 5		19
8	1019	⟨a, b \| ab=a, bba=bb⟩	Noc 5		25
8	1020	⟨a, b \| ab=a, bbb=aa⟩	Noc Com 5		9
8	1022	⟨a, b \| ab=a, bbb=ba⟩	Noc 5		4
10	8617	⟨a, b \| aa=a, abbbba=b⟩	Noc Com 5		3
11	15426	⟨a, b \| aaa=aa, abbba=b⟩	Noc Com 5

Presentation

Properties

Description

⟨a, b | aa=b, abb=1⟩

Grp Com 5

ℤ₅

1132

253

⟨a, b | aa=b, abb=a⟩

Noc Com 5

ℕ_{(5 = 1)}

254

⟨a, b | aa=b, abb=b⟩

Noc Com 5

ℕ_{(5 = 2)}

574

⟨a, b | aaa=b, aab=b⟩

Noc Com 5

ℕ_{(5 = 3)}

950

⟨a, b | ab=a, bbbb=a⟩

Noc Com 5

ℕ_{(5 = 4)}

995

⟨a, b | ab=a, aaa=bb⟩

Noc Com 5

1019

⟨a, b | ab=a, bba=bb⟩

Noc 5

1020

⟨a, b | ab=a, bbb=aa⟩

Noc Com 5

1022

⟨a, b | ab=a, bbb=ba⟩

Noc 5

8617

⟨a, b | aa=a, abbbba=b⟩

Noc Com 5

15426

⟨a, b | aaa=aa, abbba=b⟩

Noc Com 5

Isomorphic instances

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

Σ	#	Presentation	Mapping
8	587	⟨a, b \| aab=a, baa=b⟩	φ(a) = ba, φ(b) = a
8	939	⟨a, b \| ab=a, baab=b⟩	φ(a) = a, φ(b) = b
8	941	⟨a, b \| ab=a, baba=b⟩	φ(a) = a, φ(b) = b
8	945	⟨a, b \| ab=a, bbaa=b⟩	φ(a) = a, φ(b) = b
9	2975	⟨a, b \| ab=a, baabb=b⟩	φ(a) = a, φ(b) = b
9	2979	⟨a, b \| ab=a, babab=b⟩	φ(a) = a, φ(b) = b
9	2981	⟨a, b \| ab=a, babba=b⟩	φ(a) = a, φ(b) = b
9	2987	⟨a, b \| ab=a, bbaab=b⟩	φ(a) = a, φ(b) = b
9	2989	⟨a, b \| ab=a, bbaba=b⟩	φ(a) = a, φ(b) = b
9	2993	⟨a, b \| ab=a, bbbaa=b⟩	φ(a) = a, φ(b) = b
10	6353	⟨a, b \| aab=a, baaab=b⟩	φ(a) = ba, φ(b) = a
10	6355	⟨a, b \| aab=a, baaba=b⟩	φ(a) = ba, φ(b) = a
10	6359	⟨a, b \| aab=a, babaa=b⟩	φ(a) = ba, φ(b) = a
10	6365	⟨a, b \| aab=a, babbb=b⟩	φ(a) = ba, φ(b) = a
10	6367	⟨a, b \| aab=a, bbaaa=b⟩	φ(a) = ba, φ(b) = a
10	6373	⟨a, b \| aab=a, bbabb=b⟩	φ(a) = ba, φ(b) = a
10	6377	⟨a, b \| aab=a, bbbab=b⟩	φ(a) = ba, φ(b) = a
10	6379	⟨a, b \| aab=a, bbbba=b⟩	φ(a) = ba, φ(b) = a
10	8791	⟨a, b \| ab=a, baabbb=b⟩	φ(a) = a, φ(b) = b
10	8799	⟨a, b \| ab=a, bababb=b⟩	φ(a) = a, φ(b) = b
10	8803	⟨a, b \| ab=a, babbab=b⟩	φ(a) = a, φ(b) = b
10	8805	⟨a, b \| ab=a, babbba=b⟩	φ(a) = a, φ(b) = b
10	8815	⟨a, b \| ab=a, bbaabb=b⟩	φ(a) = a, φ(b) = b
10	8819	⟨a, b \| ab=a, bbabab=b⟩	φ(a) = a, φ(b) = b
10	8821	⟨a, b \| ab=a, bbabba=b⟩	φ(a) = a, φ(b) = b
10	8827	⟨a, b \| ab=a, bbbaab=b⟩	φ(a) = a, φ(b) = b
10	8829	⟨a, b \| ab=a, bbbaba=b⟩	φ(a) = a, φ(b) = b
10	8833	⟨a, b \| ab=a, bbbbaa=b⟩	φ(a) = a, φ(b) = b
11	14448	⟨a, b \| aaab=a, baabb=b⟩	φ(a) = a, φ(b) = b
11	14452	⟨a, b \| aaab=a, babab=b⟩	φ(a) = a, φ(b) = b
11	14454	⟨a, b \| aaab=a, babba=b⟩	φ(a) = a, φ(b) = b
11	14460	⟨a, b \| aaab=a, bbaab=b⟩	φ(a) = a, φ(b) = b
11	14462	⟨a, b \| aaab=a, bbaba=b⟩	φ(a) = a, φ(b) = b
11	14466	⟨a, b \| aaab=a, bbbaa=b⟩	φ(a) = a, φ(b) = b
11	14582	⟨a, b \| aaba=a, babba=b⟩	φ(a) = a, φ(b) = b
11	14590	⟨a, b \| aaba=a, bbaba=b⟩	φ(a) = a, φ(b) = b
11	14594	⟨a, b \| aaba=a, bbbaa=b⟩	φ(a) = a, φ(b) = b
11	14704	⟨a, b \| aabb=a, baabb=b⟩	φ(a) = b, φ(b) = a
11	14710	⟨a, b \| aabb=a, babba=b⟩	φ(a) = b, φ(b) = a
11	14716	⟨a, b \| aabb=a, bbaab=b⟩	φ(a) = b, φ(b) = a
11	14718	⟨a, b \| aabb=a, bbaba=b⟩	φ(a) = b, φ(b) = a
11	14722	⟨a, b \| aabb=a, bbbaa=b⟩	φ(a) = b, φ(b) = a
11	14768	⟨a, b \| abab=a, baabb=b⟩	φ(a) = b, φ(b) = a
11	14772	⟨a, b \| abab=a, babab=b⟩	φ(a) = b, φ(b) = a
11	14774	⟨a, b \| abab=a, babba=b⟩	φ(a) = b, φ(b) = a
11	14780	⟨a, b \| abab=a, bbaab=b⟩	φ(a) = b, φ(b) = a
11	14782	⟨a, b \| abab=a, bbaba=b⟩	φ(a) = b, φ(b) = a
11	14786	⟨a, b \| abab=a, bbbaa=b⟩	φ(a) = b, φ(b) = a
11	24491	⟨a, b \| ab=a, baabbbb=b⟩	φ(a) = a, φ(b) = b
11	24507	⟨a, b \| ab=a, bababbb=b⟩	φ(a) = a, φ(b) = b
11	24515	⟨a, b \| ab=a, babbabb=b⟩	φ(a) = a, φ(b) = b
11	24519	⟨a, b \| ab=a, babbbab=b⟩	φ(a) = a, φ(b) = b
11	24521	⟨a, b \| ab=a, babbbba=b⟩	φ(a) = a, φ(b) = b
11	24539	⟨a, b \| ab=a, bbaabbb=b⟩	φ(a) = a, φ(b) = b
11	24547	⟨a, b \| ab=a, bbababb=b⟩	φ(a) = a, φ(b) = b
11	24551	⟨a, b \| ab=a, bbabbab=b⟩	φ(a) = a, φ(b) = b
11	24553	⟨a, b \| ab=a, bbabbba=b⟩	φ(a) = a, φ(b) = b
11	24563	⟨a, b \| ab=a, bbbaabb=b⟩	φ(a) = a, φ(b) = b
11	24567	⟨a, b \| ab=a, bbbabab=b⟩	φ(a) = a, φ(b) = b
11	24569	⟨a, b \| ab=a, bbbabba=b⟩	φ(a) = a, φ(b) = b
11	24575	⟨a, b \| ab=a, bbbbaab=b⟩	φ(a) = a, φ(b) = b
11	24577	⟨a, b \| ab=a, bbbbaba=b⟩	φ(a) = a, φ(b) = b
11	24581	⟨a, b \| ab=a, bbbbbaa=b⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

587

⟨a, b | aab=a, baa=b⟩

φ(a) = ba, φ(b) = a

939

⟨a, b | ab=a, baab=b⟩

φ(a) = a, φ(b) = b

941

⟨a, b | ab=a, baba=b⟩

φ(a) = a, φ(b) = b

945

⟨a, b | ab=a, bbaa=b⟩

φ(a) = a, φ(b) = b

2975

⟨a, b | ab=a, baabb=b⟩

φ(a) = a, φ(b) = b

2979

⟨a, b | ab=a, babab=b⟩

φ(a) = a, φ(b) = b

2981

⟨a, b | ab=a, babba=b⟩

φ(a) = a, φ(b) = b

2987

⟨a, b | ab=a, bbaab=b⟩

φ(a) = a, φ(b) = b

2989

⟨a, b | ab=a, bbaba=b⟩

φ(a) = a, φ(b) = b

2993

⟨a, b | ab=a, bbbaa=b⟩

φ(a) = a, φ(b) = b

6353

⟨a, b | aab=a, baaab=b⟩

φ(a) = ba, φ(b) = a

6355

⟨a, b | aab=a, baaba=b⟩

φ(a) = ba, φ(b) = a

6359

⟨a, b | aab=a, babaa=b⟩

φ(a) = ba, φ(b) = a

6365

⟨a, b | aab=a, babbb=b⟩

φ(a) = ba, φ(b) = a

6367

⟨a, b | aab=a, bbaaa=b⟩

φ(a) = ba, φ(b) = a

6373

⟨a, b | aab=a, bbabb=b⟩

φ(a) = ba, φ(b) = a

6377

⟨a, b | aab=a, bbbab=b⟩

φ(a) = ba, φ(b) = a

6379

⟨a, b | aab=a, bbbba=b⟩

φ(a) = ba, φ(b) = a

8791

⟨a, b | ab=a, baabbb=b⟩

φ(a) = a, φ(b) = b

8799

⟨a, b | ab=a, bababb=b⟩

φ(a) = a, φ(b) = b

8803

⟨a, b | ab=a, babbab=b⟩

φ(a) = a, φ(b) = b

8805

⟨a, b | ab=a, babbba=b⟩

φ(a) = a, φ(b) = b

8815

⟨a, b | ab=a, bbaabb=b⟩

φ(a) = a, φ(b) = b

8819

⟨a, b | ab=a, bbabab=b⟩

φ(a) = a, φ(b) = b

8821

⟨a, b | ab=a, bbabba=b⟩

φ(a) = a, φ(b) = b

8827

⟨a, b | ab=a, bbbaab=b⟩

φ(a) = a, φ(b) = b

8829

⟨a, b | ab=a, bbbaba=b⟩

φ(a) = a, φ(b) = b

8833

⟨a, b | ab=a, bbbbaa=b⟩

φ(a) = a, φ(b) = b

14448

⟨a, b | aaab=a, baabb=b⟩

φ(a) = a, φ(b) = b

14452

⟨a, b | aaab=a, babab=b⟩

φ(a) = a, φ(b) = b

14454

⟨a, b | aaab=a, babba=b⟩

φ(a) = a, φ(b) = b

14460

⟨a, b | aaab=a, bbaab=b⟩

φ(a) = a, φ(b) = b

14462

⟨a, b | aaab=a, bbaba=b⟩

φ(a) = a, φ(b) = b

14466

⟨a, b | aaab=a, bbbaa=b⟩

φ(a) = a, φ(b) = b

14582

⟨a, b | aaba=a, babba=b⟩

φ(a) = a, φ(b) = b

14590

⟨a, b | aaba=a, bbaba=b⟩

φ(a) = a, φ(b) = b

14594

⟨a, b | aaba=a, bbbaa=b⟩

φ(a) = a, φ(b) = b

14704

⟨a, b | aabb=a, baabb=b⟩

φ(a) = b, φ(b) = a

14710

⟨a, b | aabb=a, babba=b⟩

φ(a) = b, φ(b) = a

14716

⟨a, b | aabb=a, bbaab=b⟩

φ(a) = b, φ(b) = a

14718

⟨a, b | aabb=a, bbaba=b⟩

φ(a) = b, φ(b) = a

14722

⟨a, b | aabb=a, bbbaa=b⟩

φ(a) = b, φ(b) = a

14768

⟨a, b | abab=a, baabb=b⟩

φ(a) = b, φ(b) = a

14772

⟨a, b | abab=a, babab=b⟩

φ(a) = b, φ(b) = a

14774

⟨a, b | abab=a, babba=b⟩

φ(a) = b, φ(b) = a

14780

⟨a, b | abab=a, bbaab=b⟩

φ(a) = b, φ(b) = a

14782

⟨a, b | abab=a, bbaba=b⟩

φ(a) = b, φ(b) = a

14786

⟨a, b | abab=a, bbbaa=b⟩

φ(a) = b, φ(b) = a

24491

⟨a, b | ab=a, baabbbb=b⟩

φ(a) = a, φ(b) = b

24507

⟨a, b | ab=a, bababbb=b⟩

φ(a) = a, φ(b) = b

24515

⟨a, b | ab=a, babbabb=b⟩

φ(a) = a, φ(b) = b

24519

⟨a, b | ab=a, babbbab=b⟩

φ(a) = a, φ(b) = b

24521

⟨a, b | ab=a, babbbba=b⟩

φ(a) = a, φ(b) = b

24539

⟨a, b | ab=a, bbaabbb=b⟩

φ(a) = a, φ(b) = b

24547

⟨a, b | ab=a, bbababb=b⟩

φ(a) = a, φ(b) = b

24551

⟨a, b | ab=a, bbabbab=b⟩

φ(a) = a, φ(b) = b

24553

⟨a, b | ab=a, bbabbba=b⟩

φ(a) = a, φ(b) = b

24563

⟨a, b | ab=a, bbbaabb=b⟩

φ(a) = a, φ(b) = b

24567

⟨a, b | ab=a, bbbabab=b⟩

φ(a) = a, φ(b) = b

24569

⟨a, b | ab=a, bbbabba=b⟩

φ(a) = a, φ(b) = b

24575

⟨a, b | ab=a, bbbbaab=b⟩

φ(a) = a, φ(b) = b

24577

⟨a, b | ab=a, bbbbaba=b⟩

φ(a) = a, φ(b) = b

24581

⟨a, b | ab=a, bbbbbaa=b⟩

φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

Σ	#	Presentation	Mapping
8	592	⟨a, b \| aab=b, aba=a⟩	φ(a) = ba, φ(b) = a
9	2056	⟨a, b \| aab=b, aaba=a⟩	φ(a) = a, φ(b) = b
9	2060	⟨a, b \| aab=b, abaa=a⟩	φ(a) = a, φ(b) = b
9	2068	⟨a, b \| aab=b, baaa=a⟩	φ(a) = a, φ(b) = b
10	6386	⟨a, b \| aab=b, aaaba=a⟩	φ(a) = ba, φ(b) = a
10	6390	⟨a, b \| aab=b, aabaa=a⟩	φ(a) = ba, φ(b) = a
10	6398	⟨a, b \| aab=b, abaaa=a⟩	φ(a) = ba, φ(b) = a
10	6414	⟨a, b \| aab=b, baaaa=a⟩	φ(a) = ba, φ(b) = a
10	6453	⟨a, b \| aba=a, aaabb=b⟩	φ(a) = ba, φ(b) = a
10	6457	⟨a, b \| aba=a, aabab=b⟩	φ(a) = ba, φ(b) = a
10	6463	⟨a, b \| aba=a, abaab=b⟩	φ(a) = ba, φ(b) = a
10	6473	⟨a, b \| aba=a, abbbb=b⟩	φ(a) = ba, φ(b) = a
11	14552	⟨a, b \| aaba=a, aabbb=b⟩	φ(a) = a, φ(b) = b
11	14560	⟨a, b \| aaba=a, ababb=b⟩	φ(a) = a, φ(b) = b
11	14564	⟨a, b \| aaba=a, abbab=b⟩	φ(a) = a, φ(b) = b
11	14808	⟨a, b \| abba=a, aabbb=b⟩	φ(a) = b, φ(b) = a
11	14814	⟨a, b \| abba=a, ababb=b⟩	φ(a) = b, φ(b) = a
11	14816	⟨a, b \| abba=a, abbab=b⟩	φ(a) = b, φ(b) = a
11	18964	⟨a, b \| aab=b, aaaaba=a⟩	φ(a) = a, φ(b) = b
11	18968	⟨a, b \| aab=b, aaabaa=a⟩	φ(a) = a, φ(b) = b
11	18976	⟨a, b \| aab=b, aabaaa=a⟩	φ(a) = a, φ(b) = b
11	18992	⟨a, b \| aab=b, abaaaa=a⟩	φ(a) = a, φ(b) = b
11	19024	⟨a, b \| aab=b, baaaaa=a⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

592

⟨a, b | aab=b, aba=a⟩

φ(a) = ba, φ(b) = a

2056

⟨a, b | aab=b, aaba=a⟩

φ(a) = a, φ(b) = b

2060

⟨a, b | aab=b, abaa=a⟩

φ(a) = a, φ(b) = b

2068

⟨a, b | aab=b, baaa=a⟩

φ(a) = a, φ(b) = b

6386

⟨a, b | aab=b, aaaba=a⟩

φ(a) = ba, φ(b) = a

6390

⟨a, b | aab=b, aabaa=a⟩

φ(a) = ba, φ(b) = a

6398

⟨a, b | aab=b, abaaa=a⟩

φ(a) = ba, φ(b) = a

6414

⟨a, b | aab=b, baaaa=a⟩

φ(a) = ba, φ(b) = a

6453

⟨a, b | aba=a, aaabb=b⟩

φ(a) = ba, φ(b) = a

6457

⟨a, b | aba=a, aabab=b⟩

φ(a) = ba, φ(b) = a

6463

⟨a, b | aba=a, abaab=b⟩

φ(a) = ba, φ(b) = a

6473

⟨a, b | aba=a, abbbb=b⟩

φ(a) = ba, φ(b) = a

14552

⟨a, b | aaba=a, aabbb=b⟩

φ(a) = a, φ(b) = b

14560

⟨a, b | aaba=a, ababb=b⟩

φ(a) = a, φ(b) = b

14564

⟨a, b | aaba=a, abbab=b⟩

φ(a) = a, φ(b) = b

14808

⟨a, b | abba=a, aabbb=b⟩

φ(a) = b, φ(b) = a

14814

⟨a, b | abba=a, ababb=b⟩

φ(a) = b, φ(b) = a

14816

⟨a, b | abba=a, abbab=b⟩

φ(a) = b, φ(b) = a

18964

⟨a, b | aab=b, aaaaba=a⟩

φ(a) = a, φ(b) = b

18968

⟨a, b | aab=b, aaabaa=a⟩

φ(a) = a, φ(b) = b

18976

⟨a, b | aab=b, aabaaa=a⟩

φ(a) = a, φ(b) = b

18992

⟨a, b | aab=b, abaaaa=a⟩

φ(a) = a, φ(b) = b

19024

⟨a, b | aab=b, baaaaa=a⟩

φ(a) = a, φ(b) = b

Up:	Monoid enumeration
Prev:	#267 ⟨a, b \| ab=a, baa=a⟩
Next:	#271 ⟨a, b \| ab=a, bba=a⟩

index 1, period 1	2 elements	b, a²
index 1, period 2	2 elements	a, ba

#268 ⟨a, b | ab=a, baa=b⟩

Contents

Properties

Finite structure

Cayley table

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality

Isomorphic instances

Anti-isomorphic instances