#2187 ⟨
a
,
b
,
c
|
aabb
=1,
cabc
=1⟩
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Monoid enumeration
Prev:
#2184
⟨
a
,
b
,
c
|
aabb
=1,
bcbc
=1⟩
Next:
#2189
⟨
a
,
b
,
c
|
aabc
=1,
aacb
=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
⋅
a
b
2
a
=
a
and
a
⋅ 1 =
a
, however
a
b
2
a
≠ 1
Not right cancellative, because right multiplication by
a
b
2
is not injective:
a
b
2
a
⋅
a
b
2
=
a
b
2
and 1 ⋅
a
b
2
=
a
b
2
, however
a
b
2
a
≠ 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
a
<
b
<
c
Certificate
: derivations of all rewriting rules from the defining relations.
#
Rule
Proof
1.
cab
⇒
abc
[3]
2.
a
2
b
2
⇒ 1
[1]
3.
ab
c
2
⇒ 1
[4]
# abc:aabb=1,cabc=1 abc - - cab=abc aabb=1 abcc=1