#1847 ⟨
a
,
b
,
c
|
aba
=
ac
,
ccc
=1⟩
Up:
Monoid enumeration
Prev:
#1846
⟨
a
,
b
,
c
|
aba
=
ac
,
ccb
=1⟩
Next:
#1848
⟨
a
,
b
,
c
|
aba
=
ba
,
caa
=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
⋅
c
=
aba
and
a
⋅
ba
=
aba
, however
c
≠
ba
Not right cancellative, because right multiplication by
a
is not injective:
(
ab
)
3
⋅
a
=
a
and 1 ⋅
a
=
a
, however (
ab
)
3
≠ 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(
b
) = 0,
a
<
b
; deg(
c
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
a
(
ba
)
3
⇒
a
[3]
2.
ac
⇒
aba
[1]
3.
c
3
⇒ 1
[2]
# abc:aba=ac,ccc=1 ab/c - - abababa=a ac=aba ccc=1