#3372 ⟨
a
,
b
,
c
|
ac
=
ab
,
aaa
=
a
⟩
Up:
Monoid enumeration
Prev:
#3371
⟨
a
,
b
,
c
|
ab
=
aa
,
ccc
=
c
⟩
Next:
#3373
⟨
a
,
b
,
c
|
ac
=
ab
,
aaa
=
b
⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
a
2
b
has infinite order
Not left cancellative, because left multiplication by
a
is not injective:
a
⋅
a
2
=
a
and
a
⋅ 1 =
a
, however
a
2
≠ 1
Not right cancellative, because right multiplication by
a
is not injective:
a
2
⋅
a
=
a
and 1 ⋅
a
=
a
, however
a
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.
ac
⇒
ab
[1]
2.
a
3
⇒
a
[2]
# abc:ac=ab,aaa=a abc - - ac=ab aaa=a