#12955 ⟨
a
,
b
|
aba
=
aab
,
babb
=
b
⟩
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Monoid enumeration
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#12954
⟨
a
,
b
|
aba
=
aab
,
babb
=
a
⟩
Next:
#12956
⟨
a
,
b
|
aba
=
aab
,
bbaa
=
a
⟩
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
b
is not injective:
b
⋅
a
b
2
=
b
and
b
⋅ 1 =
b
, however
a
b
2
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
bab
⋅
b
=
b
and 1 ⋅
b
=
b
, however
bab
≠ 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
⇒
aba
[1]
2.
ba
b
2
⇒
b
[2]
3.
(
ab
)
3
⇒
aba
[3]
# ab:aba=aab,babb=b ba - - aab=aba babb=b ababab=aba