#13169 ⟨
a
,
b
|
aba
=
aab
,
bab
=
bb
⟩
Up:
Monoid enumeration
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#13168
⟨
a
,
b
|
aba
=
aab
,
bab
=
ba
⟩
Next:
#13170
⟨
a
,
b
|
aba
=
aab
,
bba
=
aa
⟩
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
⋅
ab
=
b
2
and
b
⋅
b
=
b
2
, however
ab
≠
b
Not right cancellative, because right multiplication by
b
is not injective:
ba
⋅
b
=
b
2
and
b
⋅
b
=
b
2
, however
ba
≠
b
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.
bab
⇒
b
2
[2]
2.
a
2
b
⇒
aba
[1]
3.
b
3
a
⇒
b
3
[4]
4.
a
b
2
a
⇒
a
b
2
[3]
# ab:aba=aab,bab=bb ba - - bab=bb aab=aba bbba=bbb abba=abb