#40 ⟨
a
,
b
,
c
|
ac
=
ab
,
bc
=1⟩
Up:
Monoid enumeration
Prev:
#39
⟨
a
,
b
,
c
|
ac
=
ab
,
bb
=1⟩
Next:
#41
⟨
a
,
b
,
c
|
ba
=
ab
,
bc
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 6
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
b
is not injective:
b
⋅
cb
=
b
and
b
⋅ 1 =
b
, however
cb
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
cb
⋅
c
=
c
and 1 ⋅
c
=
c
, however
cb
≠ 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.
ac
⇒
ab
[1]
2.
bc
⇒ 1
[2]
# abc:ac=ab,bc=1 abc - - ac=ab bc=1