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〈a, b | aaa=1, aaba=bb〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 24 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b⁸ ⇒ b
ba ⇒ ab²
a³ ⇒ 1

Idempotents

2 elements

1
b⁷

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a²b	ab²	b³	a²b²	ab³	b⁴	a²b³	ab⁴	b⁵	a²b⁴	ab⁵	b⁶	a²b⁵	ab⁶	b⁷	a²b⁶	ab⁷	a²b⁷
1	1	a	b	a²	ab	b²	a²b	ab²	b³	a²b²	ab³	b⁴	a²b³	ab⁴	b⁵	a²b⁴	ab⁵	b⁶	a²b⁵	ab⁶	b⁷	a²b⁶	ab⁷	a²b⁷
a	a	a²	ab	1	a²b	ab²	b	a²b²	ab³	b²	a²b³	ab⁴	b³	a²b⁴	ab⁵	b⁴	a²b⁵	ab⁶	b⁵	a²b⁶	ab⁷	b⁶	a²b⁷	b⁷
b	b	ab²	b²	a²b⁴	ab³	b³	a²b⁵	ab⁴	b⁴	a²b⁶	ab⁵	b⁵	a²b⁷	ab⁶	b⁶	a²b	ab⁷	b⁷	a²b²	ab	b	a²b³	ab²	a²b⁴
a²	a²	1	a²b	a	b	a²b²	ab	b²	a²b³	ab²	b³	a²b⁴	ab³	b⁴	a²b⁵	ab⁴	b⁵	a²b⁶	ab⁵	b⁶	a²b⁷	ab⁶	b⁷	ab⁷
ab	ab	a²b²	ab²	b⁴	a²b³	ab³	b⁵	a²b⁴	ab⁴	b⁶	a²b⁵	ab⁵	b⁷	a²b⁶	ab⁶	b	a²b⁷	ab⁷	b²	a²b	ab	b³	a²b²	b⁴
b²	b²	ab⁴	b³	a²b	ab⁵	b⁴	a²b²	ab⁶	b⁵	a²b³	ab⁷	b⁶	a²b⁴	ab	b⁷	a²b⁵	ab²	b	a²b⁶	ab³	b²	a²b⁷	ab⁴	a²b
a²b	a²b	b²	a²b²	ab⁴	b³	a²b³	ab⁵	b⁴	a²b⁴	ab⁶	b⁵	a²b⁵	ab⁷	b⁶	a²b⁶	ab	b⁷	a²b⁷	ab²	b	a²b	ab³	b²	ab⁴
ab²	ab²	a²b⁴	ab³	b	a²b⁵	ab⁴	b²	a²b⁶	ab⁵	b³	a²b⁷	ab⁶	b⁴	a²b	ab⁷	b⁵	a²b²	ab	b⁶	a²b³	ab²	b⁷	a²b⁴	b
b³	b³	ab⁶	b⁴	a²b⁵	ab⁷	b⁵	a²b⁶	ab	b⁶	a²b⁷	ab²	b⁷	a²b	ab³	b	a²b²	ab⁴	b²	a²b³	ab⁵	b³	a²b⁴	ab⁶	a²b⁵
a²b²	a²b²	b⁴	a²b³	ab	b⁵	a²b⁴	ab²	b⁶	a²b⁵	ab³	b⁷	a²b⁶	ab⁴	b	a²b⁷	ab⁵	b²	a²b	ab⁶	b³	a²b²	ab⁷	b⁴	ab
ab³	ab³	a²b⁶	ab⁴	b⁵	a²b⁷	ab⁵	b⁶	a²b	ab⁶	b⁷	a²b²	ab⁷	b	a²b³	ab	b²	a²b⁴	ab²	b³	a²b⁵	ab³	b⁴	a²b⁶	b⁵
b⁴	b⁴	ab	b⁵	a²b²	ab²	b⁶	a²b³	ab³	b⁷	a²b⁴	ab⁴	b	a²b⁵	ab⁵	b²	a²b⁶	ab⁶	b³	a²b⁷	ab⁷	b⁴	a²b	ab	a²b²
a²b³	a²b³	b⁶	a²b⁴	ab⁵	b⁷	a²b⁵	ab⁶	b	a²b⁶	ab⁷	b²	a²b⁷	ab	b³	a²b	ab²	b⁴	a²b²	ab³	b⁵	a²b³	ab⁴	b⁶	ab⁵
ab⁴	ab⁴	a²b	ab⁵	b²	a²b²	ab⁶	b³	a²b³	ab⁷	b⁴	a²b⁴	ab	b⁵	a²b⁵	ab²	b⁶	a²b⁶	ab³	b⁷	a²b⁷	ab⁴	b	a²b	b²
b⁵	b⁵	ab³	b⁶	a²b⁶	ab⁴	b⁷	a²b⁷	ab⁵	b	a²b	ab⁶	b²	a²b²	ab⁷	b³	a²b³	ab	b⁴	a²b⁴	ab²	b⁵	a²b⁵	ab³	a²b⁶
a²b⁴	a²b⁴	b	a²b⁵	ab²	b²	a²b⁶	ab³	b³	a²b⁷	ab⁴	b⁴	a²b	ab⁵	b⁵	a²b²	ab⁶	b⁶	a²b³	ab⁷	b⁷	a²b⁴	ab	b	ab²
ab⁵	ab⁵	a²b³	ab⁶	b⁶	a²b⁴	ab⁷	b⁷	a²b⁵	ab	b	a²b⁶	ab²	b²	a²b⁷	ab³	b³	a²b	ab⁴	b⁴	a²b²	ab⁵	b⁵	a²b³	b⁶
b⁶	b⁶	ab⁵	b⁷	a²b³	ab⁶	b	a²b⁴	ab⁷	b²	a²b⁵	ab	b³	a²b⁶	ab²	b⁴	a²b⁷	ab³	b⁵	a²b	ab⁴	b⁶	a²b²	ab⁵	a²b³
a²b⁵	a²b⁵	b³	a²b⁶	ab⁶	b⁴	a²b⁷	ab⁷	b⁵	a²b	ab	b⁶	a²b²	ab²	b⁷	a²b³	ab³	b	a²b⁴	ab⁴	b²	a²b⁵	ab⁵	b³	ab⁶
ab⁶	ab⁶	a²b⁵	ab⁷	b³	a²b⁶	ab	b⁴	a²b⁷	ab²	b⁵	a²b	ab³	b⁶	a²b²	ab⁴	b⁷	a²b³	ab⁵	b	a²b⁴	ab⁶	b²	a²b⁵	b³
b⁷	b⁷	ab⁷	b	a²b⁷	ab	b²	a²b	ab²	b³	a²b²	ab³	b⁴	a²b³	ab⁴	b⁵	a²b⁴	ab⁵	b⁶	a²b⁵	ab⁶	b⁷	a²b⁶	ab⁷	a²b⁷
a²b⁶	a²b⁶	b⁵	a²b⁷	ab³	b⁶	a²b	ab⁴	b⁷	a²b²	ab⁵	b	a²b³	ab⁶	b²	a²b⁴	ab⁷	b³	a²b⁵	ab	b⁴	a²b⁶	ab²	b⁵	ab³
ab⁷	ab⁷	a²b⁷	ab	b⁷	a²b	ab²	b	a²b²	ab³	b²	a²b³	ab⁴	b³	a²b⁴	ab⁵	b⁴	a²b⁵	ab⁶	b⁵	a²b⁶	ab⁷	b⁶	a²b⁷	b⁷
a²b⁷	a²b⁷	b⁷	a²b	ab⁷	b	a²b²	ab	b²	a²b³	ab²	b³	a²b⁴	ab³	b⁴	a²b⁵	ab⁴	b⁵	a²b⁶	ab⁵	b⁶	a²b⁷	ab⁶	b⁷	ab⁷

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

14 unique, 72 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aab=ba, bbb=1〉	Finite non-commutative monoid with 24 elements	6 isomorphic
9	〈a, b \| aaa=1, abab=ba〉	Finite non-commutative monoid with 24 elements	2 isomorphic, 3 anti-isomorphic
10	〈a, b \| aba=bb, aabbb=1〉	Finite non-Abelian group with 24 elements	22 isomorphic
10	〈a, b \| bb=aa, aaabab=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
10	〈a, b \| aaa=1, abaab=ba〉	Finite non-commutative monoid with 24 elements	1 isomorphic
11	〈a, b \| ababa=b, abbaa=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| aaaab=1, bbbbbb=1〉	Isomorphic to ℤ₂₄	8 isomorphic
11	〈a, b \| aaabb=1, bababa=1〉	Finite non-Abelian group with 24 elements	3 isomorphic
11	〈a, b \| abba=b, aaabbb=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| abba=b, ababab=1〉	Finite non-Abelian group with 24 elements
11	〈a, b \| aba=b, aaabbbb=1〉	Finite non-Abelian group with 24 elements	7 isomorphic
11	〈a, b \| aba=a, aaaab=bb〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aba=b, aaaa=bab〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aaa=1, aabaaba=b〉	Finite non-commutative monoid with 24 elements

Other isomorphic instances

5 total

Length:	Presentation:
10	〈a, b \| aaa=1, aabb=aba〉
11	〈a, b \| aaa=1, aaaabb=ba〉
11	〈a, b \| aaa=1, aabbaa=ab〉
11	〈a, b \| aaa=1, abaaab=ba〉
11	〈a, b \| aaa=1, aaaba=abb〉