#4582 ⟨
a
,
b
,
c
|
abc
=1,
bcaa
=
b
⟩
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Monoid enumeration
Prev:
#4579
⟨
a
,
b
,
c
|
abc
=1,
bbca
=
b
⟩
Next:
#4585
⟨
a
,
b
,
c
|
abc
=1,
bcab
=
b
⟩
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
⋅
b
=
a
2
and
a
⋅
a
=
a
2
, however
b
≠
a
Not right cancellative, because right multiplication by
ac
is not injective:
aca
⋅
ac
=
ac
and 1 ⋅
ac
=
ac
, however
aca
≠ 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
c
<
a
<
b
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
⇒
a
2
[3]
2.
b
2
⇒
ba
[5]
3.
a
2
c
⇒ 1
[4]
4.
bac
⇒
bca
[6]
5.
bc
a
2
⇒
b
[2]
# abc:abc=1,bcaa=b cab - - ab=aa bb=ba aac=1 bac=bca bcaa=b