#267 ⟨a, b | ab=a, baa=a⟩

Properties

Presentation has sum-of-sides 7
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨a, b | a²=a, ab=a⟩
Not cancellative, because multiplication by a is not injective:
- b ≠ 1
- a ⋅ b = a
- a ⋅ 1 = a
- Cancellative quotient is isomorphic to ℤ₁
Grothendieck group is isomorphic to ℤ₁
Group of units is isomorphic to ℤ₁
3 Archimedian components:
- 1, a, b

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
ba ⇒ a

# ab:ab=a,baa=a ab
aa=a
ab=a
ba=a

Staircase diagram

Right Cayley graph (truncated)

Other isomorphic instances

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

168 total

Σ	#	Presentation	Mapping
8	938	⟨a, b \| ab=a, baab=a⟩	φ(a) = a, φ(b) = b
8	940	⟨a, b \| ab=a, baba=a⟩	φ(a) = a, φ(b) = b
8	1009	⟨a, b \| ab=a, baa=ab⟩	φ(a) = a, φ(b) = b
9	1614	⟨a, b \| aab=aa, baa=a⟩	φ(a) = a, φ(b) = b
9	1634	⟨a, b \| aab=ba, aba=a⟩	φ(a) = a, φ(b) = b
9	1637	⟨a, b \| aab=ba, abb=b⟩	φ(a) = b, φ(b) = a
9	1638	⟨a, b \| aab=ba, baa=a⟩	φ(a) = a, φ(b) = b
9	1667	⟨a, b \| aba=ab, abb=b⟩	φ(a) = b, φ(b) = a
9	1668	⟨a, b \| aba=ab, baa=a⟩	φ(a) = a, φ(b) = b
9	2036	⟨a, b \| aab=a, baaa=a⟩	φ(a) = a, φ(b) = b
9	2038	⟨a, b \| aab=a, baab=a⟩	φ(a) = a, φ(b) = b
9	2040	⟨a, b \| aab=a, baba=a⟩	φ(a) = a, φ(b) = b
9	2098	⟨a, b \| aba=a, baab=a⟩	φ(a) = a, φ(b) = b
9	2974	⟨a, b \| ab=a, baabb=a⟩	φ(a) = a, φ(b) = b
9	2978	⟨a, b \| ab=a, babab=a⟩	φ(a) = a, φ(b) = b
9	2980	⟨a, b \| ab=a, babba=a⟩	φ(a) = a, φ(b) = b
9	3109	⟨a, b \| ab=a, baab=ab⟩	φ(a) = a, φ(b) = b
9	3113	⟨a, b \| ab=a, baba=ab⟩	φ(a) = a, φ(b) = b
9	3181	⟨a, b \| ab=a, baa=abb⟩	φ(a) = a, φ(b) = b
10	5112	⟨a, b \| aab=aa, baab=a⟩	φ(a) = a, φ(b) = b
10	5114	⟨a, b \| aab=aa, baba=a⟩	φ(a) = a, φ(b) = b
10	5139	⟨a, b \| aab=ab, abba=b⟩	φ(a) = b, φ(b) = a
10	5160	⟨a, b \| aab=ba, aaab=a⟩	φ(a) = a, φ(b) = b
10	5162	⟨a, b \| aab=ba, aaba=a⟩	φ(a) = a, φ(b) = b
10	5166	⟨a, b \| aab=ba, abaa=a⟩	φ(a) = a, φ(b) = b
10	5169	⟨a, b \| aab=ba, abab=b⟩	φ(a) = b, φ(b) = a
10	5171	⟨a, b \| aab=ba, abba=b⟩	φ(a) = b, φ(b) = a
10	5174	⟨a, b \| aab=ba, baaa=a⟩	φ(a) = a, φ(b) = b
10	5236	⟨a, b \| aba=aa, baab=a⟩	φ(a) = a, φ(b) = b
10	5253	⟨a, b \| aba=ab, abab=b⟩	φ(a) = b, φ(b) = a
10	5255	⟨a, b \| aba=ab, abba=b⟩	φ(a) = b, φ(b) = a
10	5258	⟨a, b \| aba=ab, baaa=a⟩	φ(a) = a, φ(b) = b
10	8790	⟨a, b \| ab=a, baabbb=a⟩	φ(a) = a, φ(b) = b
10	8798	⟨a, b \| ab=a, bababb=a⟩	φ(a) = a, φ(b) = b
10	8802	⟨a, b \| ab=a, babbab=a⟩	φ(a) = a, φ(b) = b
10	8804	⟨a, b \| ab=a, babbba=a⟩	φ(a) = a, φ(b) = b
10	9061	⟨a, b \| ab=a, baabb=ab⟩	φ(a) = a, φ(b) = b
10	9069	⟨a, b \| ab=a, babab=ab⟩	φ(a) = a, φ(b) = b
10	9073	⟨a, b \| ab=a, babba=ab⟩	φ(a) = a, φ(b) = b
10	9316	⟨a, b \| ab=a, abbb=baa⟩	φ(a) = a, φ(b) = b
10	9331	⟨a, b \| ab=a, baab=abb⟩	φ(a) = a, φ(b) = b
10	9339	⟨a, b \| ab=a, baba=abb⟩	φ(a) = a, φ(b) = b
11	12226	⟨a, b \| aaab=aa, baaa=a⟩	φ(a) = a, φ(b) = b
11	12228	⟨a, b \| aaab=aa, baab=a⟩	φ(a) = a, φ(b) = b
11	12230	⟨a, b \| aaab=aa, baba=a⟩	φ(a) = a, φ(b) = b
11	12338	⟨a, b \| aaba=aa, baab=a⟩	φ(a) = a, φ(b) = b
11	12458	⟨a, b \| aabb=ab, abba=b⟩	φ(a) = b, φ(b) = a
11	12471	⟨a, b \| aabb=ba, abab=a⟩	φ(a) = a, φ(b) = b
11	12473	⟨a, b \| aabb=ba, abba=b⟩	φ(a) = b, φ(b) = a
11	12475	⟨a, b \| aabb=ba, abbb=b⟩	φ(a) = b, φ(b) = a
11	12519	⟨a, b \| abab=ab, abba=b⟩	φ(a) = b, φ(b) = a
11	12532	⟨a, b \| abab=ba, abba=a⟩	φ(a) = a, φ(b) = b
11	12533	⟨a, b \| abab=ba, abba=b⟩	φ(a) = b, φ(b) = a
11	12535	⟨a, b \| abab=ba, abbb=b⟩	φ(a) = b, φ(b) = a
11	12536	⟨a, b \| abab=ba, baaa=a⟩	φ(a) = a, φ(b) = b
11	12538	⟨a, b \| abab=ba, baba=a⟩	φ(a) = a, φ(b) = b
11	12540	⟨a, b \| abab=ba, babb=b⟩	φ(a) = b, φ(b) = a
11	12541	⟨a, b \| abab=ba, bbaa=a⟩	φ(a) = a, φ(b) = b
11	12543	⟨a, b \| abab=ba, bbab=b⟩	φ(a) = b, φ(b) = a
11	12569	⟨a, b \| abba=ab, abbb=b⟩	φ(a) = b, φ(b) = a
11	12570	⟨a, b \| abba=ab, baaa=a⟩	φ(a) = a, φ(b) = b
11	12572	⟨a, b \| abba=ab, baab=a⟩	φ(a) = a, φ(b) = b
11	12574	⟨a, b \| abba=ab, baba=a⟩	φ(a) = a, φ(b) = b
11	12589	⟨a, b \| abba=ba, baba=b⟩	φ(a) = b, φ(b) = a
11	12630	⟨a, b \| baba=ab, bbaa=a⟩	φ(a) = a, φ(b) = b
11	14441	⟨a, b \| aaab=a, baaaa=a⟩	φ(a) = a, φ(b) = b
11	14707	⟨a, b \| aabb=a, babab=a⟩	φ(a) = a, φ(b) = b
11	14767	⟨a, b \| abab=a, baabb=a⟩	φ(a) = a, φ(b) = b
11	14771	⟨a, b \| abab=a, babab=a⟩	φ(a) = a, φ(b) = b
11	14773	⟨a, b \| abab=a, babba=a⟩	φ(a) = a, φ(b) = b
11	14825	⟨a, b \| abba=a, babab=a⟩	φ(a) = a, φ(b) = b
11	14840	⟨a, b \| abba=b, aaabb=b⟩	φ(a) = b, φ(b) = a
11	14844	⟨a, b \| abba=b, aabab=b⟩	φ(a) = b, φ(b) = a
11	14846	⟨a, b \| abba=b, aabba=b⟩	φ(a) = b, φ(b) = a
11	14850	⟨a, b \| abba=b, abaab=b⟩	φ(a) = b, φ(b) = a
11	14852	⟨a, b \| abba=b, ababa=b⟩	φ(a) = b, φ(b) = a
11	14862	⟨a, b \| abba=b, baaab=b⟩	φ(a) = b, φ(b) = a
11	15583	⟨a, b \| aab=aa, baabb=a⟩	φ(a) = a, φ(b) = b
11	15589	⟨a, b \| aab=aa, babba=a⟩	φ(a) = a, φ(b) = b
11	15622	⟨a, b \| aab=ab, aabba=b⟩	φ(a) = b, φ(b) = a
11	15630	⟨a, b \| aab=ab, ababa=b⟩	φ(a) = b, φ(b) = a
11	15675	⟨a, b \| aab=ba, aaaab=a⟩	φ(a) = a, φ(b) = b
11	15677	⟨a, b \| aab=ba, aaaba=a⟩	φ(a) = a, φ(b) = b
11	15680	⟨a, b \| aab=ba, aaabb=b⟩	φ(a) = b, φ(b) = a
11	15681	⟨a, b \| aab=ba, aabaa=a⟩	φ(a) = a, φ(b) = b
11	15689	⟨a, b \| aab=ba, abaaa=a⟩	φ(a) = a, φ(b) = b
11	15692	⟨a, b \| aab=ba, abaab=b⟩	φ(a) = b, φ(b) = a
11	15694	⟨a, b \| aab=ba, ababa=b⟩	φ(a) = b, φ(b) = a
11	15698	⟨a, b \| aab=ba, abbaa=b⟩	φ(a) = b, φ(b) = a
11	15705	⟨a, b \| aab=ba, baaaa=a⟩	φ(a) = a, φ(b) = b
11	15833	⟨a, b \| aba=aa, babab=a⟩	φ(a) = a, φ(b) = b
11	15860	⟨a, b \| aba=ab, abaab=b⟩	φ(a) = b, φ(b) = a
11	15862	⟨a, b \| aba=ab, ababa=b⟩	φ(a) = b, φ(b) = a
11	15866	⟨a, b \| aba=ab, abbaa=b⟩	φ(a) = b, φ(b) = a
11	15873	⟨a, b \| aba=ab, baaaa=a⟩	φ(a) = a, φ(b) = b
11	18898	⟨a, b \| aab=a, baaaab=a⟩	φ(a) = a, φ(b) = b
11	18900	⟨a, b \| aab=a, baaaba=a⟩	φ(a) = a, φ(b) = b
11	18902	⟨a, b \| aab=a, baaabb=a⟩	φ(a) = a, φ(b) = b
11	18904	⟨a, b \| aab=a, baabaa=a⟩	φ(a) = a, φ(b) = b
11	18906	⟨a, b \| aab=a, baabab=a⟩	φ(a) = a, φ(b) = b
11	18908	⟨a, b \| aab=a, baabba=a⟩	φ(a) = a, φ(b) = b
11	18912	⟨a, b \| aab=a, babaaa=a⟩	φ(a) = a, φ(b) = b
11	18914	⟨a, b \| aab=a, babaab=a⟩	φ(a) = a, φ(b) = b
11	18916	⟨a, b \| aab=a, bababa=a⟩	φ(a) = a, φ(b) = b
11	18920	⟨a, b \| aab=a, babbaa=a⟩	φ(a) = a, φ(b) = b
11	19140	⟨a, b \| aba=a, baaaab=a⟩	φ(a) = a, φ(b) = b
11	19144	⟨a, b \| aba=a, baabab=a⟩	φ(a) = a, φ(b) = b
11	19390	⟨a, b \| aab=a, aaabb=ba⟩	φ(a) = a, φ(b) = b
11	19398	⟨a, b \| aab=a, aabab=ba⟩	φ(a) = a, φ(b) = b
11	19402	⟨a, b \| aab=a, aabba=ba⟩	φ(a) = a, φ(b) = b
11	19406	⟨a, b \| aab=a, aabbb=ba⟩	φ(a) = a, φ(b) = b
11	19414	⟨a, b \| aab=a, abaab=ba⟩	φ(a) = a, φ(b) = b
11	19418	⟨a, b \| aab=a, ababa=ba⟩	φ(a) = a, φ(b) = b
11	19422	⟨a, b \| aab=a, ababb=ba⟩	φ(a) = a, φ(b) = b
11	19426	⟨a, b \| aab=a, abbaa=ba⟩	φ(a) = a, φ(b) = b
11	19430	⟨a, b \| aab=a, abbab=ba⟩	φ(a) = a, φ(b) = b
11	19434	⟨a, b \| aab=a, abbba=ba⟩	φ(a) = a, φ(b) = b
11	19440	⟨a, b \| aab=a, baaaa=aa⟩	φ(a) = a, φ(b) = b
11	19444	⟨a, b \| aab=a, baaab=aa⟩	φ(a) = a, φ(b) = b
11	19445	⟨a, b \| aab=a, baaab=ab⟩	φ(a) = a, φ(b) = b
11	19448	⟨a, b \| aab=a, baaba=aa⟩	φ(a) = a, φ(b) = b
11	19449	⟨a, b \| aab=a, baaba=ab⟩	φ(a) = a, φ(b) = b
11	19453	⟨a, b \| aab=a, baabb=ab⟩	φ(a) = a, φ(b) = b
11	19456	⟨a, b \| aab=a, babaa=aa⟩	φ(a) = a, φ(b) = b
11	19457	⟨a, b \| aab=a, babaa=ab⟩	φ(a) = a, φ(b) = b
11	19461	⟨a, b \| aab=a, babab=ab⟩	φ(a) = a, φ(b) = b
11	19465	⟨a, b \| aab=a, babba=ab⟩	φ(a) = a, φ(b) = b
11	19652	⟨a, b \| aba=a, aabab=ba⟩	φ(a) = a, φ(b) = b
11	19664	⟨a, b \| aba=a, abaab=ba⟩	φ(a) = a, φ(b) = b
11	19675	⟨a, b \| aba=a, abbab=ba⟩	φ(a) = a, φ(b) = b
11	19932	⟨a, b \| aab=a, aaab=baa⟩	φ(a) = a, φ(b) = b
11	19940	⟨a, b \| aab=a, aaba=baa⟩	φ(a) = a, φ(b) = b
11	19948	⟨a, b \| aab=a, aabb=baa⟩	φ(a) = a, φ(b) = b
11	19949	⟨a, b \| aab=a, aabb=bab⟩	φ(a) = a, φ(b) = b
11	19956	⟨a, b \| aab=a, abaa=baa⟩	φ(a) = a, φ(b) = b
11	19964	⟨a, b \| aab=a, abab=baa⟩	φ(a) = a, φ(b) = b
11	19965	⟨a, b \| aab=a, abab=bab⟩	φ(a) = a, φ(b) = b
11	19972	⟨a, b \| aab=a, abba=baa⟩	φ(a) = a, φ(b) = b
11	19973	⟨a, b \| aab=a, abba=bab⟩	φ(a) = a, φ(b) = b
11	19981	⟨a, b \| aab=a, abbb=bab⟩	φ(a) = a, φ(b) = b
11	19984	⟨a, b \| aab=a, baaa=aaa⟩	φ(a) = a, φ(b) = b
11	19985	⟨a, b \| aab=a, baaa=aab⟩	φ(a) = a, φ(b) = b
11	19986	⟨a, b \| aab=a, baaa=aba⟩	φ(a) = a, φ(b) = b
11	19993	⟨a, b \| aab=a, baab=aab⟩	φ(a) = a, φ(b) = b
11	19994	⟨a, b \| aab=a, baab=aba⟩	φ(a) = a, φ(b) = b
11	19995	⟨a, b \| aab=a, baab=abb⟩	φ(a) = a, φ(b) = b
11	20001	⟨a, b \| aab=a, baba=aab⟩	φ(a) = a, φ(b) = b
11	20002	⟨a, b \| aab=a, baba=aba⟩	φ(a) = a, φ(b) = b
11	20003	⟨a, b \| aab=a, baba=abb⟩	φ(a) = a, φ(b) = b
11	20011	⟨a, b \| aab=a, babb=abb⟩	φ(a) = a, φ(b) = b
11	20210	⟨a, b \| aba=a, abab=baa⟩	φ(a) = a, φ(b) = b
11	20218	⟨a, b \| aba=a, abba=bab⟩	φ(a) = a, φ(b) = b
11	20230	⟨a, b \| aba=a, baab=aba⟩	φ(a) = a, φ(b) = b
11	24490	⟨a, b \| ab=a, baabbbb=a⟩	φ(a) = a, φ(b) = b
11	24506	⟨a, b \| ab=a, bababbb=a⟩	φ(a) = a, φ(b) = b
11	24514	⟨a, b \| ab=a, babbabb=a⟩	φ(a) = a, φ(b) = b
11	24518	⟨a, b \| ab=a, babbbab=a⟩	φ(a) = a, φ(b) = b
11	24520	⟨a, b \| ab=a, babbbba=a⟩	φ(a) = a, φ(b) = b
11	25017	⟨a, b \| ab=a, baabbb=ab⟩	φ(a) = a, φ(b) = b
11	25033	⟨a, b \| ab=a, bababb=ab⟩	φ(a) = a, φ(b) = b
11	25041	⟨a, b \| ab=a, babbab=ab⟩	φ(a) = a, φ(b) = b
11	25045	⟨a, b \| ab=a, babbba=ab⟩	φ(a) = a, φ(b) = b
11	25528	⟨a, b \| ab=a, abbbb=baa⟩	φ(a) = a, φ(b) = b
11	25559	⟨a, b \| ab=a, baabb=abb⟩	φ(a) = a, φ(b) = b
11	25575	⟨a, b \| ab=a, babab=abb⟩	φ(a) = a, φ(b) = b
11	25583	⟨a, b \| ab=a, babba=abb⟩	φ(a) = a, φ(b) = b
11	25835	⟨a, b \| ab=a, baab=abbb⟩	φ(a) = a, φ(b) = b
11	25844	⟨a, b \| ab=a, baba=abbb⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#265 ⟨a, b \| ab=a, abb=a⟩
Next:	#268 ⟨a, b \| ab=a, baa=b⟩