#3019 ⟨
a
,
b
|
aabaaabaabb
=1⟩
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Monoid enumeration
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#3015
⟨
a
,
b
|
aabaaaabbab
=1⟩
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#3021
⟨
a
,
b
|
aabaaababab
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅ (
ab
a
2
)
2
b
2
a
=
a
and
a
⋅ 1 =
a
, however (
ab
a
2
)
2
b
2
a
≠ 1
Not right cancellative, because right multiplication by (
ab
a
2
)
2
b
2
is not injective:
(
ab
a
2
)
2
b
2
a
⋅ (
ab
a
2
)
2
b
2
= (
ab
a
2
)
2
b
2
and 1 ⋅ (
ab
a
2
)
2
b
2
= (
ab
a
2
)
2
b
2
, however (
ab
a
2
)
2
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
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
(
ab
a
2
)
2
b
2
⇒ 1
[1]
# ab:aabaaabaabb=1 ab - - aabaaabaabb=1