#2225 ⟨
a
,
b
,
c
|
aabc
=1,
cacb
=1⟩
Up:
Monoid enumeration
Prev:
#2219
⟨
a
,
b
,
c
|
aabc
=1,
caab
=1⟩
Next:
#2228
⟨
a
,
b
,
c
|
aabc
=1,
cbac
=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
⋅
abca
=
a
and
a
⋅ 1 =
a
, however
abca
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ac
⋅
b
=
a
2
b
and
a
2
⋅
b
=
a
2
b
, however
ac
≠
a
2
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.
acb
⇒
a
2
b
[3]
2.
a
2
bc
⇒ 1
[1]
3.
c
a
2
b
⇒ 1
[4]
# abc:aabc=1,cacb=1 abc - - acb=aab aabc=1 caab=1