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〈a, b | aaa=1, aabaaba=b〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 24 elements

Completion parameters

Reduction order: left-to-right ab

Complete rewriting system

a³ ⇒ 1
ba²b ⇒ aba²
(ba)² ⇒ a²b²
bab² ⇒ ab²a
b²a² ⇒ (ab)²
b²ab ⇒ a²ba
b³a ⇒ ab³
b⁴ ⇒ b

Idempotents

2 elements

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	b²	a²b	aba	ab²	ba²	bab	b²a	b³	a²ba	a²b²	aba²	(ab)²	ab²a	ab³	a²ba²	a(ab)²	a²b²a	a²b³
1	1	a	b	a²	ab	ba	b²	a²b	aba	ab²	ba²	bab	b²a	b³	a²ba	a²b²	aba²	(ab)²	ab²a	ab³	a²ba²	a(ab)²	a²b²a	a²b³
a	a	a²	ab	1	a²b	aba	ab²	b	a²ba	a²b²	aba²	(ab)²	ab²a	ab³	ba	b²	a²ba²	a(ab)²	a²b²a	a²b³	ba²	bab	b²a	b³
b	b	ba	b²	ba²	bab	b²a	b³	aba²	a²b²	ab²a	(ab)²	a²ba	ab³	b	ab	a²ba²	a²b²a	a²b³	a(ab)²	ba	aba	ab²	a²b	ba²
a²	a²	1	a²b	a	b	a²ba	a²b²	ab	ba	b²	a²ba²	a(ab)²	a²b²a	a²b³	aba	ab²	ba²	bab	b²a	b³	aba²	(ab)²	ab²a	ab³
ab	ab	aba	ab²	aba²	(ab)²	ab²a	ab³	a²ba²	b²	a²b²a	a(ab)²	ba	a²b³	ab	a²b	ba²	b²a	b³	bab	aba	a²ba	a²b²	b	aba²
ba	ba	ba²	bab	b	aba²	a²b²	ab²a	b²	ab	a²ba²	a²b²a	a²b³	a(ab)²	ba	b²a	b³	aba	ab²	a²b	ba²	(ab)²	a²ba	ab³	b
b²	b²	b²a	b³	(ab)²	a²ba	ab³	b	a²b²a	a²ba²	a(ab)²	a²b³	ab	ba	b²	bab	aba	a²b	ba²	ab²	b²a	a²b²	ab²a	aba²	(ab)²
a²b	a²b	a²ba	a²b²	a²ba²	a(ab)²	a²b²a	a²b³	ba²	ab²	b²a	bab	aba	b³	a²b	b	aba²	ab²a	ab³	(ab)²	a²ba	ba	b²	ab	a²ba²
aba	aba	aba²	(ab)²	ab	a²ba²	b²	a²b²a	ab²	a²b	ba²	b²a	b³	bab	aba	ab²a	ab³	a²ba	a²b²	b	aba²	a(ab)²	ba	a²b³	ab
ab²	ab²	ab²a	ab³	a(ab)²	ba	a²b³	ab	b²a	ba²	bab	b³	a²b	aba	ab²	(ab)²	a²ba	b	aba²	a²b²	ab²a	b²	a²b²a	a²ba²	a(ab)²
ba²	ba²	b	aba²	ba	b²	ab	a²ba²	bab	b²a	b³	aba	ab²	a²b	ba²	a²b²	ab²a	(ab)²	a²ba	ab³	b	a²b²a	a²b³	a(ab)²	ba
bab	bab	a²b²	ab²a	a²b²a	a²b³	a(ab)²	ba	aba	b³	a²b	ab²	b²a	ba²	bab	aba²	(ab)²	ab³	b	a²ba	a²b²	ab	a²ba²	b²	a²b²a
b²a	b²a	(ab)²	a²ba	b²	a²b²a	a²ba²	a(ab)²	b³	bab	aba	a²b	ba²	ab²	b²a	ab³	b	a²b²	ab²a	aba²	(ab)²	a²b³	ab	ba	b²
b³	b³	ab³	b	a²b³	ab	ba	b²	a²b	aba	ab²	ba²	bab	b²a	b³	a²ba	a²b²	aba²	(ab)²	ab²a	ab³	a²ba²	a(ab)²	a²b²a	a²b³
a²ba	a²ba	a²ba²	a(ab)²	a²b	ba²	ab²	b²a	a²b²	b	aba²	ab²a	ab³	(ab)²	a²ba	a²b²a	a²b³	ba	b²	ab	a²ba²	bab	aba	b³	a²b
a²b²	a²b²	a²b²a	a²b³	bab	aba	b³	a²b	ab²a	aba²	(ab)²	ab³	b	a²ba	a²b²	a(ab)²	ba	ab	a²ba²	b²	a²b²a	ab²	b²a	ba²	bab
aba²	aba²	ab	a²ba²	aba	ab²	a²b	ba²	(ab)²	ab²a	ab³	a²ba	a²b²	b	aba²	b²	a²b²a	a(ab)²	ba	a²b³	ab	b²a	b³	bab	aba
(ab)²	(ab)²	b²	a²b²a	b²a	b³	bab	aba	a²ba	ab³	b	a²b²	ab²a	aba²	(ab)²	a²ba²	a(ab)²	a²b³	ab	ba	b²	a²b	ba²	ab²	b²a
ab²a	ab²a	a(ab)²	ba	ab²	b²a	ba²	bab	ab³	(ab)²	a²ba	b	aba²	a²b²	ab²a	a²b³	ab	b²	a²b²a	a²ba²	a(ab)²	b³	a²b	aba	ab²
ab³	ab³	a²b³	ab	b³	a²b	aba	ab²	b	a²ba	a²b²	aba²	(ab)²	ab²a	ab³	ba	b²	a²ba²	a(ab)²	a²b²a	a²b³	ba²	bab	b²a	b³
a²ba²	a²ba²	a²b	ba²	a²ba	a²b²	b	aba²	a(ab)²	a²b²a	a²b³	ba	b²	ab	a²ba²	ab²	b²a	bab	aba	b³	a²b	ab²a	ab³	(ab)²	a²ba
a(ab)²	a(ab)²	ab²	b²a	ab²a	ab³	(ab)²	a²ba	ba	a²b³	ab	b²	a²b²a	a²ba²	a(ab)²	ba²	bab	b³	a²b	aba	ab²	b	aba²	a²b²	ab²a
a²b²a	a²b²a	bab	aba	a²b²	ab²a	aba²	(ab)²	a²b³	a(ab)²	ba	ab	a²ba²	b²	a²b²a	b³	a²b	ab²	b²a	ba²	bab	ab³	b	a²ba	a²b²
a²b³	a²b³	b³	a²b	ab³	b	a²ba	a²b²	ab	ba	b²	a²ba²	a(ab)²	a²b²a	a²b³	aba	ab²	ba²	bab	b²a	b³	aba²	(ab)²	ab²a	ab³

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

14 unique, 77 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
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