#3107 ⟨
a
,
b
,
c
|
ab
=
aa
,
bcbc
=1⟩
Up:
Monoid enumeration
Prev:
#3104
⟨
a
,
b
,
c
|
ab
=
aa
,
bcac
=1⟩
Next:
#3109
⟨
a
,
b
,
c
|
ab
=
aa
,
bccb
=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
⋅
b
=
a
2
and
a
⋅
a
=
a
2
, however
b
≠
a
Not right cancellative, because right multiplication by
cbc
is not injective:
(
cb
)
2
⋅
cbc
=
cbc
and 1 ⋅
cbc
=
cbc
, however (
cb
)
2
≠ 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.
ab
⇒
a
2
[1]
2.
(
bc
)
2
⇒ 1
[2]
3.
a
2
cbc
⇒
a
[3]
# abc:ab=aa,bcbc=1 abc - - ab=aa bcbc=1 aacbc=a