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〈a, b | aba=a, aaaab=bb〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 24 elements

Completion parameters

Reduction order: left-to-right a/b

Complete rewriting system

a⁹ ⇒ a⁴
ba⁴ ⇒ a⁸
aba ⇒ a
b² ⇒ a⁴b

Idempotents

5 elements

1
a⁵
ba
ab
a⁶b

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
1	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a	a	a²	ab	a³	a²b	a	a⁴	a³b	a²	ab	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a³b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
b	b	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a²	a²	a³	a²b	a⁴	a³b	a²	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ab	ab	a	a⁵b	a²	ab	a⁵	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
ba	ba	ba²	bab	ba³	ba²b	ba	a⁸	ba³b	ba²	bab	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a³	a³	a⁴	a³b	a⁵	a⁴b	a³	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a²b	a²b	a²	a⁶b	a³	a²b	a⁶	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
ba²	ba²	ba³	ba²b	a⁸	ba³b	ba²	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
bab	bab	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a³b	a³b	a³	a⁷b	a⁴	a³b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba³	ba³	a⁸	ba³b	a⁴	a⁸b	ba³	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba²b	ba²b	ba²	a⁵b	ba³	ba²b	a⁵	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁵	a⁵	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁴b	a⁴b	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
ba³b	ba³b	ba³	a⁶b	a⁸	ba³b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁶	a⁶	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁵b	a⁵b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁷	a⁷	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
a⁶b	a⁶b	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁸	a⁸	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a⁷b	a⁷b	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁸b	a⁸b	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b

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=b, aaaa=bab〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aaa=1, aabaaba=b〉	Finite non-commutative monoid with 24 elements