#5878 ⟨
a
,
b
|
aabb
=1,
abaab
=
a
⟩
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Monoid enumeration
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#5873
⟨
a
,
b
|
aabb
=1,
aabab
=
b
⟩
Next:
#5900
⟨
a
,
b
|
abab
=1,
aaabb
=
a
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
b
2
a
=
a
and
a
⋅ 1 =
a
, however
a
b
2
a
≠ 1
Not right cancellative, because right multiplication by
a
b
2
is not injective:
a
b
2
a
⋅
a
b
2
=
a
b
2
and 1 ⋅
a
b
2
=
a
b
2
, however
a
b
2
a
≠ 1
Rewriting system
Format:
Pretty
Plain
Word:
Enter a word above to compute its normal form. Tips:
Lowercase letters stand for generators.
Spaces are ignored.
Numbers repeat the previous letter, e.g.
b90
.
Strategy:
Leftmost
Rightmost
Result:
1
1
Reduction order:
Left-to-right shortlex with
b
<
a
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
2
b
2
⇒ 1
[1]
2.
a
3
b
⇒
ab
a
2
[3]
3.
ab
a
2
b
⇒
a
[2]
# ab:aabb=1,abaab=a ba - - aabb=1 aaab=abaa abaab=a