#1848 ⟨
a
,
b
,
c
|
aba
=
ba
,
caa
=1⟩
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Monoid enumeration
Prev:
#1847
⟨
a
,
b
,
c
|
aba
=
ac
,
ccc
=1⟩
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#1849
⟨
a
,
b
,
c
|
aba
=
bb
,
abc
=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
c
is not injective:
c
⋅
a
2
c
=
c
and
c
⋅ 1 =
c
, however
a
2
c
≠ 1
Not right cancellative, because right multiplication by
d
is not injective:
a
⋅
d
=
d
and 1 ⋅
d
=
d
, however
a
≠ 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
=
ba
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.
ad
⇒
d
[5]
2.
ba
⇒
d
[3]
3.
bd
⇒
d
2
[6]
4.
cd
⇒
d
[7]
5.
c
a
2
⇒ 1
[2]
# abc:aba=ba,caa=1 adbc ba=d morph:2/1 ad=d ba=d bd=dd cd=d caa=1