#13153 ⟨
a
,
b
|
aba
=
aaa
,
bab
=
ab
⟩
Up:
Monoid enumeration
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#13152
⟨
a
,
b
|
aba
=
aaa
,
bab
=
aa
⟩
Next:
#13154
⟨
a
,
b
|
aba
=
aaa
,
bab
=
bb
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
=
aba
and
a
⋅
ba
=
aba
, however
a
2
≠
ba
Not right cancellative, because right multiplication by
b
is not injective:
ba
⋅
b
=
ab
and
a
⋅
b
=
ab
, however
ba
≠
a
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
⇒
ab
[2]
2.
a
3
⇒
aba
[1]
3.
b
a
2
b
⇒
a
2
b
[3]
4.
a
2
ba
⇒
ab
a
2
[4]
# ab:aba=aaa,bab=ab ba - - bab=ab aaa=aba baab=aab aaba=abaa