#1592 ⟨
a
,
b
,
c
|
aaa
=
bc
,
cab
=1⟩
Up:
Monoid enumeration
Prev:
#1590
⟨
a
,
b
,
c
|
aaa
=
bc
,
bbc
=1⟩
Next:
#1593
⟨
a
,
b
,
c
|
aaa
=
bc
,
cbb
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
acba
has infinite order
Not left cancellative, because left multiplication by
c
is not injective:
c
⋅ (
bc
)
4
=
c
and
c
⋅ 1 =
c
, however (
bc
)
4
≠ 1
Not right cancellative, because right multiplication by
b
is not injective:
a
⋅
b
=
b
(
cb
)
3
and (
bc
)
3
⋅
b
=
b
(
cb
)
3
, however
a
≠ (
bc
)
3
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(
b
) = deg(
c
) = 0,
b
<
c
; deg(
a
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
(
cb
)
4
⇒ 1
[8]
2.
ab
⇒
b
(
cb
)
3
[9]
3.
ca
⇒
c
(
bc
)
3
[7]
4.
a
3
⇒
bc
[1]
# abc:aaa=bc,cab=1 bc/a - - cbcbcbcb=1 ab=bcbcbcb ca=cbcbcbc aaa=bc