#157 ⟨
a
,
b
|
ab
=
aa
,
ba
=
a
⟩
Up:
Monoid enumeration
Prev:
#151
⟨
a
,
b
|
ba
=
ab
,
aaa
=1⟩
Next:
#158
⟨
a
,
b
|
ab
=
aa
,
ba
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
=
a
2
and
a
⋅
a
=
a
2
, however
b
≠
a
Not right cancellative, because right multiplication by
a
is not injective:
b
⋅
a
=
a
and 1 ⋅
a
=
a
, however
b
≠ 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.
ab
⇒
a
2
[1]
2.
ba
⇒
a
[2]
3.
a
3
⇒
a
2
[3]
# ab:ab=aa,ba=a ab - - ab=aa ba=a aaa=aa