#1630 ⟨
a
,
b
|
aab
=
ab
,
bba
=
a
⟩
Up:
Monoid enumeration
Prev:
#1629
⟨
a
,
b
|
aab
=
ab
,
bab
=
b
⟩
Next:
#1631
⟨
a
,
b
|
aab
=
ab
,
bba
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 9
Infinite non-cancellative non-commutative monoid
Element
ba
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
=
a
2
and
a
⋅
a
=
a
2
, however
a
2
≠
a
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.
a
3
⇒
a
2
[3]
2.
a
2
b
⇒
ab
[1]
3.
b
2
a
⇒
a
[2]
# ab:aab=ab,bba=a ab - - aaa=aa aab=ab bba=a