#3205 ⟨
a
,
b
|
abaabaabbab
=1⟩
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Monoid enumeration
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#3195
⟨
a
,
b
|
abaaabbabab
=1⟩
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#3222
⟨
a
,
b
|
abaabbbabab
=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
⋅
b
(
a
2
b
)
2
(
ba
)
2
=
a
and
a
⋅ 1 =
a
, however
b
(
a
2
b
)
2
(
ba
)
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ab
(
a
2
b
)
2
⋅
b
= (
a
2
b
)
2
bab
and (
a
2
b
)
2
ba
⋅
b
= (
a
2
b
)
2
bab
, however
ab
(
a
2
b
)
2
≠ (
a
2
b
)
2
ba
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
(
b
a
2
)
2
b
2
⇒ (
a
2
b
)
2
bab
[2]
2.
(
a
2
b
)
2
b
(
ab
)
2
⇒ 1
[3]
# ab:abaabaabbab=1 ab - - abaabaabb=aabaabbab aabaabbabab=1