#50 ⟨
a
,
b
,
c
|
bb
=
ac
,
ca
=1⟩
Up:
Monoid enumeration
Prev:
#49
⟨
a
,
b
,
c
|
bb
=
ac
,
bc
=1⟩
Next:
#53
⟨
a
,
b
,
c
|
bc
=
ac
,
cc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅
ac
=
c
and
c
⋅ 1 =
c
, however
ac
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
ac
⋅
a
=
a
and 1 ⋅
a
=
a
, however
ac
≠ 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.
b
2
⇒
ac
[1]
2.
ca
⇒ 1
[2]
3.
bac
⇒
acb
[3]
4.
acba
⇒
ba
[4]
# abc:bb=ac,ca=1 abc - - bb=ac ca=1 bac=acb acba=ba