#6172 ⟨
a
,
b
,
c
|
aa
=1,
abbccc
=1⟩
Up:
Monoid enumeration
Prev:
#6171
⟨
a
,
b
,
c
|
aa
=1,
abbccb
=1⟩
Next:
#6176
⟨
a
,
b
,
c
|
aa
=1,
abcabc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element
b
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅ (
b
c
3
b
)
2
=
b
and
b
⋅ 1 =
b
, however (
b
c
3
b
)
2
≠ 1
Not right cancellative, because right multiplication by
b
c
3
b
2
c
3
is not injective:
(
b
c
3
b
)
2
⋅
b
c
3
b
2
c
3
=
b
c
3
b
2
c
3
and 1 ⋅
b
c
3
b
2
c
3
=
b
c
3
b
2
c
3
, however (
b
c
3
b
)
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 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.
(
b
2
c
3
)
2
⇒ 1
[4]
2.
a
⇒
b
2
c
3
[3]
# abc:aa=1,abbccc=1 bc/a - - bbcccbbccc=1 a=bbccc