#1807 ⟨
a
,
b
,
c
|
aba
=
ab
,
abc
=1⟩
Up:
Monoid enumeration
Prev:
#1806
⟨
a
,
b
,
c
|
aba
=
aa
,
ccc
=1⟩
Next:
#1809
⟨
a
,
b
,
c
|
aba
=
ab
,
acb
=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
d
is not injective:
d
⋅
a
=
d
and
d
⋅ 1 =
d
, however
a
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
cd
⋅
c
=
c
and 1 ⋅
c
=
c
, however
cd
≠ 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
Auxiliary generators:
d
=
ab
Reduction order:
Left-to-right shortlex with
a
<
d
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
ab
⇒
d
[3]
2.
da
⇒
d
[5]
3.
db
⇒
d
2
[7]
4.
dc
⇒ 1
[6]
# abc:aba=ab,abc=1 adbc ab=d morph:2/1 ab=d da=d db=dd dc=1