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〈a, b | aaa=1, abab=ba〉

Monoid presentation of length 9

Properties

Finite non-commutative monoid with 24 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a³ ⇒ 1
bab ⇒ a²ba
ab² ⇒ (ba²)²
(a²b)² ⇒ b²a
b²a²b ⇒ aba²
b⁴ ⇒ b

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a²b	aba	ba²	b²a	b³	a²ba	aba²	ba²b	b²a²	b³a	a²ba²	aba²b	ba²ba	b³a²	(aba)²	(ba²)²	a(ba²)²
1	1	a	b	a²	ab	ba	b²	a²b	aba	ba²	b²a	b³	a²ba	aba²	ba²b	b²a²	b³a	a²ba²	aba²b	ba²ba	b³a²	(aba)²	(ba²)²	a(ba²)²
a	a	a²	ab	1	a²b	aba	(ba²)²	b	a²ba	aba²	ba²b	b³a	ba	a²ba²	aba²b	ba²ba	b³a²	ba²	b²a	(aba)²	b³	b²a²	a(ba²)²	b²
b	b	ba	b²	ba²	a²ba	b²a	b³	ba²b	a²ba²	b²a²	b³a	b	ba²ba	a²b	aba²	b³a²	ba	(ba²)²	a(ba²)²	ab	ba²	aba²b	aba	(aba)²
a²	a²	1	a²b	a	b	a²ba	a(ba²)²	ab	ba	a²ba²	aba²b	b³a²	aba	ba²	b²a	(aba)²	b³	aba²	ba²b	b²a²	b³a	ba²ba	b²	(ba²)²
ab	ab	aba	(ba²)²	aba²	ba	ba²b	b³a	aba²b	ba²	ba²ba	b³a²	ab	(aba)²	b	a²ba²	b³	aba	a(ba²)²	b²	a²b	aba²	b²a	a²ba	b²a²
ba	ba	ba²	a²ba	b	ba²b	a²ba²	aba	b²	ba²ba	a²b	aba²	ba	b²a	(ba²)²	a(ba²)²	ab	ba²	b²a²	b³a	aba²b	b	b³a²	(aba)²	b³
b²	b²	b²a	b³	b²a²	ba²ba	b³a	b	aba²	(ba²)²	b³a²	ba	b²	ab	ba²b	a²b	ba²	b²a	aba	(aba)²	a²ba	b²a²	a(ba²)²	a²ba²	aba²b
a²b	a²b	a²ba	a(ba²)²	a²ba²	aba	aba²b	b³a²	b²a	aba²	(aba)²	b³	a²b	b²a²	ab	ba²	b³a	a²ba	b²	(ba²)²	b	a²ba²	ba²b	ba	ba²ba
aba	aba	aba²	ba	ab	aba²b	ba²	a²ba	(ba²)²	(aba)²	b	a²ba²	aba	ba²b	a(ba²)²	b²	a²b	aba²	ba²ba	b³a²	b²a	ab	b³	b²a²	b³a
ba²	ba²	b	ba²b	ba	b²	ba²ba	(aba)²	a²ba	b²a	(ba²)²	a(ba²)²	ba²	a²ba²	b²a²	b³a	aba²b	b	a²b	aba²	b³a²	ba	ab	b³	aba
b²a	b²a	b²a²	ba²ba	b²	aba²	(ba²)²	a²ba²	b³	ab	ba²b	a²b	b²a	b³a	aba	(aba)²	a²ba	b²a²	b³a²	ba	a(ba²)²	b²	ba²	aba²b	b
b³	b³	b³a	b	b³a²	ab	ba	b²	a²b	aba	ba²	b²a	b³	a²ba	aba²	ba²b	b²a²	b³a	a²ba²	aba²b	ba²ba	b³a²	(aba)²	(ba²)²	a(ba²)²
a²ba	a²ba	a²ba²	aba	a²b	b²a	aba²	ba	a(ba²)²	b²a²	ab	ba²	a²ba	aba²b	b²	(ba²)²	b	a²ba²	(aba)²	b³	ba²b	a²b	b³a	ba²ba	b³a²
aba²	aba²	ab	aba²b	aba	(ba²)²	(aba)²	b²a²	ba	ba²b	a(ba²)²	b²	aba²	ba²	ba²ba	b³a²	b²a	ab	b	a²ba²	b³	aba	a²b	b³a	a²ba
ba²b	ba²b	ba²ba	(aba)²	(ba²)²	a²ba²	a(ba²)²	ba²	b³a	a²b	aba²b	b	ba²b	b³a²	a²ba	b²a²	ba	ba²ba	b³	aba	b²	(ba²)²	aba²	b²a	ab
b²a²	b²a²	b²	aba²	b²a	b³	ab	aba²b	ba²ba	b³a	aba	(aba)²	b²a²	(ba²)²	b³a²	ba	a(ba²)²	b²	ba²b	a²b	ba²	b²a	a²ba	b	a²ba²
b³a	b³a	b³a²	ab	b³	a²b	aba	(ba²)²	b	a²ba	aba²	ba²b	b³a	ba	a²ba²	aba²b	ba²ba	b³a²	ba²	b²a	(aba)²	b³	b²a²	a(ba²)²	b²
a²ba²	a²ba²	a²b	b²a	a²ba	a(ba²)²	b²a²	ba²ba	aba	aba²b	b²	(ba²)²	a²ba²	aba²	(aba)²	b³	ba²b	a²b	ab	ba²	b³a	a²ba	b	b³a²	ba
aba²b	aba²b	(aba)²	b²a²	a(ba²)²	ba²	b²	aba²	b³a²	b	b²a	ab	aba²b	b³	ba	ba²ba	aba	(aba)²	b³a	a²ba	(ba²)²	a(ba²)²	a²ba²	ba²b	a²b
ba²ba	ba²ba	(ba²)²	a²ba²	ba²b	b³a	a²b	b²a	(aba)²	b³a²	a²ba	b²a²	ba²ba	a(ba²)²	b³	aba	b²	(ba²)²	aba²b	b	aba²	ba²b	ba	ab	ba²
b³a²	b³a²	b³	a²b	b³a	b	a²ba	a(ba²)²	ab	ba	a²ba²	aba²b	b³a²	aba	ba²	b²a	(aba)²	b³	aba²	ba²b	b²a²	b³a	ba²ba	b²	(ba²)²
(aba)²	(aba)²	a(ba²)²	ba²	aba²b	b³a²	b	ba²b	b²a²	b³	ba	ba²ba	(aba)²	b²	b³a	a²ba	(ba²)²	a(ba²)²	b²a	ab	a²ba²	aba²b	aba	a²b	aba²
(ba²)²	(ba²)²	ba²b	b³a	ba²ba	(aba)²	b³a²	ab	a²ba²	a(ba²)²	b³	aba	(ba²)²	a²b	aba²b	b	aba²	ba²b	a²ba	b²a²	ba	ba²ba	b²	ba²	b²a
a(ba²)²	a(ba²)²	aba²b	b³a²	(aba)²	b²a²	b³	a²b	ba²	b²	b³a	a²ba	a(ba²)²	b	b²a	ab	a²ba²	aba²b	ba	ba²ba	aba	(aba)²	(ba²)²	aba²	ba²b

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, aaba=bb〉	Finite non-commutative monoid with 24 elements	5 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

2 total

Length:	Presentation:
10	〈a, b \| aaa=1, aaba=bab〉
11	〈a, b \| aaa=1, aabab=aba〉

Other anti-isomorphic instances

3 total

Length:	Presentation:
10	〈a, b \| aaa=1, aababa=b〉
11	〈a, b \| aaa=1, aabab=baa〉
11	〈a, b \| aaa=1, ababa=aab〉