#3181 ⟨a, b, c | ba=ab, aaca=1⟩

Contents

Properties
Commutative structure
Rewriting system
Other submonoids of same group

Properties

Sum of relation sides is 8
Isomorphic to ℕ ⊕ ℤ
Infinite cancellative commutative monoid
Element a has infinite order

Commutative structure

Gröbner basis: ⟨a, b, c | a³c=1⟩
Enveloping group is isomorphic to ℤ ⊕ ℤ
Inverses of generators:
- a^-1 = a²c
- c^-1 = a³
Group of units is isomorphic to ℤ
2 Archimedian components:
- 1, b

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 < c < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ca ⇒ ac	[6]
2.	ba ⇒ ab	[1]
3.	bc ⇒ cb	[8]
4.	a³c ⇒ 1	[9]

# abc:ba=ab,aaca=1 acb - -
ca=ac
ba=ab
bc=cb
aaac=1

Other submonoids of same group

3 unique, 14 total

Σ	#	Presentation	Properties	Description	φ
6	83	⟨a, b, c \| abc=1, cba=1⟩	Grp Com Inf	ℤ ⊕ ℤ	4
6	159	⟨a, b, c \| ab=c, ba=c⟩	Can Com Inf	ℕ ⊕ ℕ	5
7	461	⟨a, b, c \| ba=ab, aca=1⟩	Can Com Inf	ℕ ⊕ ℤ	2