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