#1554 ⟨
a
,
b
,
c
|
aaa
=
ab
,
bac
=1⟩
Up:
Monoid enumeration
Prev:
#1553
⟨
a
,
b
,
c
|
aaa
=
ab
,
acc
=1⟩
Next:
#1555
⟨
a
,
b
,
c
|
aaa
=
ab
,
bbc
=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
b
is not injective:
b
⋅
acb
=
b
and
b
⋅ 1 =
b
, however
acb
≠ 1
Not right cancellative, because right multiplication by
ac
is not injective:
acb
⋅
ac
=
ac
and 1 ⋅
ac
=
ac
, however
acb
≠ 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 recursive path with deg(
a
) = deg(
c
) = 0,
a
<
c
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
4
c
⇒
a
[3]
2.
bac
⇒ 1
[2]
3.
ab
⇒
a
3
[1]
# abc:aaa=ab,bac=1 ac/b - - aaaac=a bac=1 ab=aaa