#243 ⟨a, b | aa=a, bab=a⟩

Properties

Presentation has sum-of-sides 7
Infinite non-cancellative commutative monoid
Commutative Gröbner basis: ⟨a, b | a²=a, ab²=a⟩
Not cancellative, because multiplication by ab is not injective:
- a ≠ 1
- ab ⋅ a = ab
- ab ⋅ 1 = ab
- Cancellative quotient is isomorphic to ℤ₂
Grothendieck group is isomorphic to ℤ₂
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
ba ⇒ ab
ab² ⇒ a

# ab:aa=a,bab=a ab
aa=a
ba=ab
abb=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.

100 total

Σ	#	Presentation	Mapping
8	894	⟨a, b \| aa=a, baab=a⟩	φ(a) = a, φ(b) = b
8	966	⟨a, b \| aa=a, bab=aa⟩	φ(a) = a, φ(b) = b
9	1628	⟨a, b \| aab=ab, bab=a⟩	φ(a) = a, φ(b) = b
9	1640	⟨a, b \| aab=ba, bab=a⟩	φ(a) = a, φ(b) = b
9	1647	⟨a, b \| aab=bb, aba=b⟩	φ(a) = b, φ(b) = a
9	1670	⟨a, b \| aba=ab, bab=a⟩	φ(a) = a, φ(b) = b
9	2111	⟨a, b \| aba=b, aabb=b⟩	φ(a) = b, φ(b) = a
9	2113	⟨a, b \| aba=b, abab=b⟩	φ(a) = b, φ(b) = a
9	2115	⟨a, b \| aba=b, abba=b⟩	φ(a) = b, φ(b) = a
9	2119	⟨a, b \| aba=b, baab=b⟩	φ(a) = b, φ(b) = a
9	2884	⟨a, b \| aa=a, baaab=a⟩	φ(a) = a, φ(b) = b
9	3026	⟨a, b \| aa=a, baab=aa⟩	φ(a) = a, φ(b) = b
9	3144	⟨a, b \| aa=a, bab=aaa⟩	φ(a) = a, φ(b) = b
10	5036	⟨a, b \| aaa=aa, baab=a⟩	φ(a) = a, φ(b) = b
10	5144	⟨a, b \| aab=ab, baab=a⟩	φ(a) = a, φ(b) = b
10	5164	⟨a, b \| aab=ba, aabb=a⟩	φ(a) = a, φ(b) = b
10	5168	⟨a, b \| aab=ba, abab=a⟩	φ(a) = a, φ(b) = b
10	5170	⟨a, b \| aab=ba, abba=a⟩	φ(a) = a, φ(b) = b
10	5178	⟨a, b \| aab=ba, baba=a⟩	φ(a) = a, φ(b) = b
10	5182	⟨a, b \| aab=ba, bbaa=a⟩	φ(a) = a, φ(b) = b
10	5260	⟨a, b \| aba=ab, baab=a⟩	φ(a) = a, φ(b) = b
10	5262	⟨a, b \| aba=ab, baba=a⟩	φ(a) = a, φ(b) = b
10	8620	⟨a, b \| aa=a, baaaab=a⟩	φ(a) = a, φ(b) = b
10	8892	⟨a, b \| aa=a, baaab=aa⟩	φ(a) = a, φ(b) = b
10	9116	⟨a, b \| aa=a, aaaa=bab⟩	φ(a) = a, φ(b) = b
10	9164	⟨a, b \| aa=a, baab=aaa⟩	φ(a) = a, φ(b) = b
11	12273	⟨a, b \| aaab=ba, aabb=b⟩	φ(a) = b, φ(b) = a
11	12277	⟨a, b \| aaab=ba, abab=b⟩	φ(a) = b, φ(b) = a
11	12279	⟨a, b \| aaab=ba, abba=b⟩	φ(a) = b, φ(b) = a
11	12353	⟨a, b \| aaba=ab, aabb=b⟩	φ(a) = b, φ(b) = a
11	12357	⟨a, b \| aaba=ab, abab=b⟩	φ(a) = b, φ(b) = a
11	12359	⟨a, b \| aaba=ab, abba=b⟩	φ(a) = b, φ(b) = a
11	12504	⟨a, b \| abab=aa, baab=a⟩	φ(a) = a, φ(b) = b
11	12506	⟨a, b \| abab=aa, baba=a⟩	φ(a) = a, φ(b) = b
11	12599	⟨a, b \| abba=bb, baba=b⟩	φ(a) = b, φ(b) = a
11	12603	⟨a, b \| abba=bb, bbaa=b⟩	φ(a) = b, φ(b) = a
11	14699	⟨a, b \| aabb=a, baaab=a⟩	φ(a) = a, φ(b) = b
11	14763	⟨a, b \| abab=a, baaab=a⟩	φ(a) = a, φ(b) = b
11	14765	⟨a, b \| abab=a, baaba=a⟩	φ(a) = a, φ(b) = b
11	14769	⟨a, b \| abab=a, babaa=a⟩	φ(a) = a, φ(b) = b
11	14821	⟨a, b \| abba=a, baaab=a⟩	φ(a) = a, φ(b) = b
11	14848	⟨a, b \| abba=b, aabbb=b⟩	φ(a) = b, φ(b) = a
11	14856	⟨a, b \| abba=b, abbab=b⟩	φ(a) = b, φ(b) = a
11	14858	⟨a, b \| abba=b, abbba=b⟩	φ(a) = b, φ(b) = a
11	14864	⟨a, b \| abba=b, baabb=b⟩	φ(a) = b, φ(b) = a
11	15429	⟨a, b \| aaa=aa, baaab=a⟩	φ(a) = a, φ(b) = b
11	15643	⟨a, b \| aab=ab, baaab=a⟩	φ(a) = a, φ(b) = b
11	15679	⟨a, b \| aab=ba, aaabb=a⟩	φ(a) = a, φ(b) = b
11	15685	⟨a, b \| aab=ba, aabba=a⟩	φ(a) = a, φ(b) = b
11	15691	⟨a, b \| aab=ba, abaab=a⟩	φ(a) = a, φ(b) = b
11	15697	⟨a, b \| aab=ba, abbaa=a⟩	φ(a) = a, φ(b) = b
11	15707	⟨a, b \| aab=ba, baaab=a⟩	φ(a) = a, φ(b) = b
11	15709	⟨a, b \| aab=ba, baaba=a⟩	φ(a) = a, φ(b) = b
11	15721	⟨a, b \| aab=ba, bbaaa=a⟩	φ(a) = a, φ(b) = b
11	15875	⟨a, b \| aba=ab, baaab=a⟩	φ(a) = a, φ(b) = b
11	15877	⟨a, b \| aba=ab, baaba=a⟩	φ(a) = a, φ(b) = b
11	15881	⟨a, b \| aba=ab, babaa=a⟩	φ(a) = a, φ(b) = b
11	19167	⟨a, b \| aba=b, aaaabb=b⟩	φ(a) = b, φ(b) = a
11	19171	⟨a, b \| aba=b, aaabab=b⟩	φ(a) = b, φ(b) = a
11	19173	⟨a, b \| aba=b, aaabba=b⟩	φ(a) = b, φ(b) = a
11	19177	⟨a, b \| aba=b, aabaab=b⟩	φ(a) = b, φ(b) = a
11	19179	⟨a, b \| aba=b, aababa=b⟩	φ(a) = b, φ(b) = a
11	19183	⟨a, b \| aba=b, aabbaa=b⟩	φ(a) = b, φ(b) = a
11	19191	⟨a, b \| aba=b, abaaab=b⟩	φ(a) = b, φ(b) = a
11	19193	⟨a, b \| aba=b, abaaba=b⟩	φ(a) = b, φ(b) = a
11	19213	⟨a, b \| aba=b, baaaab=b⟩	φ(a) = b, φ(b) = a
11	19710	⟨a, b \| aba=b, aaaab=bb⟩	φ(a) = b, φ(b) = a
11	19714	⟨a, b \| aba=b, aaaba=bb⟩	φ(a) = b, φ(b) = a
11	19716	⟨a, b \| aba=b, aaabb=ab⟩	φ(a) = b, φ(b) = a
11	19717	⟨a, b \| aba=b, aaabb=ba⟩	φ(a) = b, φ(b) = a
11	19721	⟨a, b \| aba=b, aabaa=bb⟩	φ(a) = b, φ(b) = a
11	19723	⟨a, b \| aba=b, aabab=ab⟩	φ(a) = b, φ(b) = a
11	19724	⟨a, b \| aba=b, aabab=ba⟩	φ(a) = b, φ(b) = a
11	19727	⟨a, b \| aba=b, aabba=ab⟩	φ(a) = b, φ(b) = a
11	19728	⟨a, b \| aba=b, aabba=ba⟩	φ(a) = b, φ(b) = a
11	19735	⟨a, b \| aba=b, abaab=ab⟩	φ(a) = b, φ(b) = a
11	19736	⟨a, b \| aba=b, abaab=ba⟩	φ(a) = b, φ(b) = a
11	19739	⟨a, b \| aba=b, ababa=ab⟩	φ(a) = b, φ(b) = a
11	19757	⟨a, b \| aba=b, baaab=ab⟩	φ(a) = b, φ(b) = a
11	20257	⟨a, b \| aba=b, aaab=abb⟩	φ(a) = b, φ(b) = a
11	20259	⟨a, b \| aba=b, aaab=bab⟩	φ(a) = b, φ(b) = a
11	20260	⟨a, b \| aba=b, aaab=bba⟩	φ(a) = b, φ(b) = a
11	20265	⟨a, b \| aba=b, aaba=abb⟩	φ(a) = b, φ(b) = a
11	20267	⟨a, b \| aba=b, aaba=bab⟩	φ(a) = b, φ(b) = a
11	20268	⟨a, b \| aba=b, aaba=bba⟩	φ(a) = b, φ(b) = a
11	20271	⟨a, b \| aba=b, aabb=aab⟩	φ(a) = b, φ(b) = a
11	20272	⟨a, b \| aba=b, aabb=aba⟩	φ(a) = b, φ(b) = a
11	20274	⟨a, b \| aba=b, aabb=baa⟩	φ(a) = b, φ(b) = a
11	20279	⟨a, b \| aba=b, abab=aab⟩	φ(a) = b, φ(b) = a
11	20280	⟨a, b \| aba=b, abab=aba⟩	φ(a) = b, φ(b) = a
11	20282	⟨a, b \| aba=b, abab=baa⟩	φ(a) = b, φ(b) = a
11	20287	⟨a, b \| aba=b, abba=aab⟩	φ(a) = b, φ(b) = a
11	20288	⟨a, b \| aba=b, abba=aba⟩	φ(a) = b, φ(b) = a
11	20301	⟨a, b \| aba=b, baab=aab⟩	φ(a) = b, φ(b) = a
11	20302	⟨a, b \| aba=b, baab=aba⟩	φ(a) = b, φ(b) = a
11	24148	⟨a, b \| aa=a, baaaaab=a⟩	φ(a) = a, φ(b) = b
11	24688	⟨a, b \| aa=a, baaaab=aa⟩	φ(a) = a, φ(b) = b
11	25120	⟨a, b \| aa=a, aaaaa=bab⟩	φ(a) = a, φ(b) = b
11	25220	⟨a, b \| aa=a, baaab=aaa⟩	φ(a) = a, φ(b) = b
11	25690	⟨a, b \| aa=a, baab=aaaa⟩	φ(a) = a, φ(b) = b

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