#4146 ⟨
a
,
b
|
aba
=
aab
,
bba
=
a
⟩
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Monoid enumeration
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#4145
⟨
a
,
b
|
aba
=
aab
,
bab
=
b
⟩
Next:
#4147
⟨
a
,
b
|
aba
=
aab
,
bba
=
b
⟩
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
⋅
ba
=
a
2
b
and
a
⋅
ab
=
a
2
b
, however
ba
≠
ab
Not right cancellative, because right multiplication by
a
is not injective:
b
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
b
2
≠ 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.
aba
⇒
a
2
b
[1]
2.
b
2
a
⇒
a
[2]
3.
a
3
b
2
⇒
a
3
[3]
# ab:aba=aab,bba=a ab - - aba=aab bba=a aaabb=aaa