#9 ⟨
a
,
b
,
c
|
ab
=
a
,
bc
=1⟩
Up:
Monoid enumeration
Prev:
#8
⟨
a
,
b
,
c
|
ab
=
a
,
ac
=1⟩
Next:
#11
⟨
a
,
b
,
c
|
ab
=
a
,
cb
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 5
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
b
=
a
and
a
⋅ 1 =
a
, however
b
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
cb
⋅
c
=
c
and 1 ⋅
c
=
c
, however
cb
≠ 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
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
⇒
a
[1]
2.
ac
⇒
a
[3]
3.
bc
⇒ 1
[2]
# abc:ab=a,bc=1 abc - - ab=a ac=a bc=1