#5100 ⟨
a
,
b
,
c
|
ab
=
c
,
cacbc
=1⟩
Up:
Monoid enumeration
Prev:
#5051
⟨
a
,
b
,
c
|
ab
=
c
,
acccc
=1⟩
Next:
#5116
⟨
a
,
b
,
c
|
aa
=
a
,
abac
=
a
⟩
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
⋅
ab
(
ba
)
3
=
a
and
a
⋅ 1 =
a
, however
ab
(
ba
)
3
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
ab
a
2
b
⋅
b
=
a
2
b
2
ab
and
a
2
b
2
a
⋅
b
=
a
2
b
2
ab
, however
ab
a
2
b
≠
a
2
b
2
a
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(
b
) = 0,
a
<
b
; deg(
c
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ab
a
2
b
2
⇒
a
2
b
2
ab
[3]
2.
a
2
b
2
(
ab
)
2
⇒ 1
[4]
3.
c
⇒
ab
[1]
# abc:ab=c,cacbc=1 ab/c - - abaabb=aabbab aabbabab=1 c=ab