#3211 ⟨
a
,
b
,
c
|
ba
=
ac
,
abbc
=1⟩
Up:
Monoid enumeration
Prev:
#3210
⟨
a
,
b
,
c
|
ba
=
ac
,
abbb
=1⟩
Next:
#3213
⟨
a
,
b
,
c
|
ba
=
ac
,
abcb
=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
⋅
b
2
ca
=
a
and
a
⋅ 1 =
a
, however
b
2
ca
≠ 1
Not right cancellative, because right multiplication by
b
2
c
is not injective:
b
2
ca
⋅
b
2
c
=
b
2
c
and 1 ⋅
b
2
c
=
b
2
c
, however
b
2
ca
≠ 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 shortlex with
b
<
a
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
ac
⇒
ba
[1]
2.
a
b
2
c
⇒ 1
[2]
# abc:ba=ac,abbc=1 bac - - ac=ba abbc=1