#460 ⟨
a
,
b
,
c
|
ba
=
ab
,
abc
=1⟩
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Monoid enumeration
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#459
⟨
a
,
b
,
c
|
ba
=
ab
,
aac
=1⟩
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#461
⟨
a
,
b
,
c
|
ba
=
ab
,
aca
=1⟩
Contents
Properties
Rewriting system
Properties
Sum of relation sides is 7
Infinite non-cancellative non-commutative monoid
Element
a
has infinite order
Not left cancellative, because left multiplication by
d
is not injective:
d
⋅
cd
=
d
and
d
⋅ 1 =
d
, however
cd
≠ 1
Not right cancellative, because right multiplication by
c
is not injective:
cd
⋅
c
=
c
and 1 ⋅
c
=
c
, however
cd
≠ 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
Auxiliary generators:
d
=
ba
Reduction order:
Left-to-right recursive path with deg(
a
) = deg(
c
) = deg(
d
) = 0,
a
<
c
<
d
; deg(
b
) = 1
Certificate
: derivations of all rewriting rules from the defining relations.
Morphocompletion
: how the auxiliary generators were found.
#
Rule
Proof
1.
da
⇒
ad
[6]
2.
dc
⇒ 1
[5]
3.
ba
⇒
d
[3]
4.
ab
⇒
d
[4]
5.
db
⇒
bd
[7]
# abc:ba=ab,abc=1 acd/b ba=d morph:2/1 da=ad dc=1 ba=d ab=d db=bd