#1849 ⟨
a
,
b
,
c
|
aba
=
bb
,
abc
=1⟩
Up:
Monoid enumeration
Prev:
#1848
⟨
a
,
b
,
c
|
aba
=
ba
,
caa
=1⟩
Next:
#1850
⟨
a
,
b
,
c
|
aba
=
bb
,
aca
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
bca
=
a
and
a
⋅ 1 =
a
, however
bca
≠ 1
Not right cancellative, because right multiplication by
bc
is not injective:
bca
⋅
bc
=
bc
and 1 ⋅
bc
=
bc
, however
bca
≠ 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.
aba
⇒
b
2
[1]
2.
abc
⇒ 1
[2]
3.
b
3
a
⇒
a
b
3
[3]
4.
b
3
c
⇒
ab
[4]
# abc:aba=bb,abc=1 abc - - aba=bb abc=1 bbba=abbb bbbc=ab