#1111 ⟨
a
,
b
|
abaaba
=
aaa
⟩
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Monoid enumeration
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#1110
⟨
a
,
b
|
abaaab
=
bbb
⟩
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#1112
⟨
a
,
b
|
abaaba
=
aab
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 9
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
ba
=
a
3
and
a
⋅
a
2
=
a
3
, however
b
a
2
ba
≠
a
2
Not right cancellative, because right multiplication by
a
is not injective:
ab
a
2
b
⋅
a
=
a
3
and
a
2
⋅
a
=
a
3
, however
ab
a
2
b
≠
a
2
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.
ab
a
4
⇒
a
4
ba
[2]
2.
(
aba
)
2
⇒
a
3
[1]
# ab:abaaba=aaa ab - - abaaaa=aaaaba abaaba=aaa