#11 ⟨a, b | aa=1, aaa=1⟩

Up:Monoids with two generators and two relations
Prev:#9 ⟨a, b | ab=a, bb=1⟩
Next:#16 ⟨a, b | aa=1, bbb=1⟩

Properties

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. a ⇒ 1 
# ab:aa=1,aaa=1 ab
a=1

Staircase diagram

Right Cayley graph (truncated)

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

98 total

Σ#PresentationMapping
635⟨a, b | aa=a, aaa=1⟩φ(a) = 1, φ(b) = b
7161⟨a, b | ba=ab, bb=a⟩φ(a) = bb, φ(b) = b
7163⟨a, b | aaa=1, aaaa=1⟩φ(a) = 1, φ(b) = b
7199⟨a, b | aa=a, aaaa=1⟩φ(a) = 1, φ(b) = b
7283⟨a, b | aa=1, aaaaa=1⟩φ(a) = 1, φ(b) = b
7319⟨a, b | aa=1, aaaa=a⟩φ(a) = 1, φ(b) = b
7355⟨a, b | aa=1, aaa=aa⟩φ(a) = 1, φ(b) = b
8647⟨a, b | ba=ab, aaa=b⟩φ(a) = b, φ(b) = bbb
8664⟨a, b | aaa=1, aaaaa=1⟩φ(a) = 1, φ(b) = b
8808⟨a, b | aa=a, aaaaa=1⟩φ(a) = 1, φ(b) = b
8973⟨a, b | aa=b, aaa=ab⟩φ(a) = b, φ(b) = bb
8978⟨a, b | aa=b, aab=bb⟩φ(a) = b, φ(b) = bb
8981⟨a, b | aa=b, aba=bb⟩φ(a) = b, φ(b) = bb
91440⟨a, b | aaa=aa, aaaa=1⟩φ(a) = 1, φ(b) = b
91700⟨a, b | aaaa=1, aaaaa=1⟩φ(a) = 1, φ(b) = b
91836⟨a, b | aaa=a, aaaaa=1⟩φ(a) = 1, φ(b) = b
92209⟨a, b | ba=ab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
92416⟨a, b | aaa=1, aaaaa=a⟩φ(a) = 1, φ(b) = b
92560⟨a, b | aaa=1, aaaa=aa⟩φ(a) = 1, φ(b) = b
92720⟨a, b | aa=a, aaaaaa=1⟩φ(a) = 1, φ(b) = b
93038⟨a, b | aa=b, aaaa=bb⟩φ(a) = b, φ(b) = bb
93156⟨a, b | aa=b, aba=aab⟩φ(a) = b, φ(b) = bb
93160⟨a, b | aa=b, baa=aab⟩φ(a) = b, φ(b) = bb
93165⟨a, b | aa=b, bab=abb⟩φ(a) = b, φ(b) = bb
93166⟨a, b | aa=b, bba=abb⟩φ(a) = b, φ(b) = bb
93200⟨a, b | aa=1, aaaaaaa=1⟩φ(a) = 1, φ(b) = b
93336⟨a, b | aa=1, aaaaaa=a⟩φ(a) = 1, φ(b) = b
93472⟨a, b | aa=1, aaaaa=aa⟩φ(a) = 1, φ(b) = b
93608⟨a, b | aa=1, aaaa=aaa⟩φ(a) = 1, φ(b) = b
104148⟨a, b | aba=aab, bbb=a⟩φ(a) = bbb, φ(b) = b
104176⟨a, b | baa=aab, bbb=a⟩φ(a) = bbb, φ(b) = b
104194⟨a, b | bab=abb, bbb=a⟩φ(a) = bbb, φ(b) = b
104198⟨a, b | bba=abb, bbb=a⟩φ(a) = bbb, φ(b) = b
104342⟨a, b | aaaa=a, aaaaa=1⟩φ(a) = 1, φ(b) = b
104750⟨a, b | aaa=aa, aaaaa=1⟩φ(a) = 1, φ(b) = b
106563⟨a, b | aaa=b, aaaa=ab⟩φ(a) = b, φ(b) = bbb
106568⟨a, b | aaa=b, aaab=bb⟩φ(a) = b, φ(b) = bbb
106572⟨a, b | aaa=b, aaba=bb⟩φ(a) = b, φ(b) = bbb
107015⟨a, b | ba=ab, aaaaa=b⟩φ(a) = b, φ(b) = bbbbb
107208⟨a, b | aaa=1, aaaaaaa=1⟩φ(a) = 1, φ(b) = b
107480⟨a, b | aaa=1, aaaaaa=a⟩φ(a) = 1, φ(b) = b
108024⟨a, b | aaa=1, aaaa=aaa⟩φ(a) = 1, φ(b) = b
108296⟨a, b | aa=a, aaaaaaa=1⟩φ(a) = 1, φ(b) = b
109185⟨a, b | aa=b, aaaa=aab⟩φ(a) = b, φ(b) = bb
109186⟨a, b | aa=b, aaaa=aba⟩φ(a) = b, φ(b) = bb
109193⟨a, b | aa=b, aaab=abb⟩φ(a) = b, φ(b) = bb
109195⟨a, b | aa=b, aaab=bab⟩φ(a) = b, φ(b) = bb
109196⟨a, b | aa=b, aaab=bba⟩φ(a) = b, φ(b) = bb
109201⟨a, b | aa=b, aaba=abb⟩φ(a) = b, φ(b) = bb
109203⟨a, b | aa=b, aaba=bab⟩φ(a) = b, φ(b) = bb
109204⟨a, b | aa=b, aaba=bba⟩φ(a) = b, φ(b) = bb
109213⟨a, b | aa=b, aabb=bbb⟩φ(a) = b, φ(b) = bb
109221⟨a, b | aa=b, abab=bbb⟩φ(a) = b, φ(b) = bb
109227⟨a, b | aa=b, abba=bbb⟩φ(a) = b, φ(b) = bb
109241⟨a, b | aa=b, baab=bbb⟩φ(a) = b, φ(b) = bb
1111620⟨a, b | aaaa=aa, aaaaa=1⟩φ(a) = 1, φ(b) = b
1112933⟨a, b | aba=aab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1112962⟨a, b | aba=aab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1113045⟨a, b | baa=aab, aaaa=b⟩φ(a) = b, φ(b) = bbbb
1113062⟨a, b | baa=aab, bbbb=a⟩φ(a) = bbbb, φ(b) = b
1113273⟨a, b | aaaaa=1, aaaaaa=1⟩φ(a) = 1, φ(b) = b
1114873⟨a, b | aaa=aa, aaaaaa=1⟩φ(a) = 1, φ(b) = b
1116560⟨a, b | aaaa=1, aaaaaaa=1⟩φ(a) = 1, φ(b) = b
1117088⟨a, b | aaaa=1, aaaaaa=a⟩φ(a) = 1, φ(b) = b
1117616⟨a, b | aaaa=1, aaaaa=aa⟩φ(a) = 1, φ(b) = b
1118144⟨a, b | aaa=a, aaaaaaa=1⟩φ(a) = 1, φ(b) = b
1120649⟨a, b | ba=ab, aaaaaa=b⟩φ(a) = b, φ(b) = bbbbbb
1121120⟨a, b | aaa=1, aaaaaaaa=1⟩φ(a) = 1, φ(b) = b
1122192⟨a, b | aaa=1, aaaaaa=aa⟩φ(a) = 1, φ(b) = b
1122720⟨a, b | aaa=1, aaaaa=aaa⟩φ(a) = 1, φ(b) = b
1123516⟨a, b | aa=a, aaaaaaaa=1⟩φ(a) = 1, φ(b) = b
1125263⟨a, b | aa=b, aaaaa=abb⟩φ(a) = b, φ(b) = bb
1125264⟨a, b | aa=b, aaaaa=bab⟩φ(a) = b, φ(b) = bb
1125273⟨a, b | aa=b, aaaab=bbb⟩φ(a) = b, φ(b) = bb
1125281⟨a, b | aa=b, aaaba=bbb⟩φ(a) = b, φ(b) = bb
1125295⟨a, b | aa=b, aabaa=bbb⟩φ(a) = b, φ(b) = bb
1125728⟨a, b | aa=b, aaba=aaab⟩φ(a) = b, φ(b) = bb
1125732⟨a, b | aa=b, abaa=aaab⟩φ(a) = b, φ(b) = bb
1125733⟨a, b | aa=b, abaa=aaba⟩φ(a) = b, φ(b) = bb
1125738⟨a, b | aa=b, abab=aabb⟩φ(a) = b, φ(b) = bb
1125743⟨a, b | aa=b, abba=aabb⟩φ(a) = b, φ(b) = bb
1125744⟨a, b | aa=b, abba=abab⟩φ(a) = b, φ(b) = bb
1125752⟨a, b | aa=b, baaa=aaab⟩φ(a) = b, φ(b) = bb
1125759⟨a, b | aa=b, baab=aabb⟩φ(a) = b, φ(b) = bb
1125760⟨a, b | aa=b, baab=abab⟩φ(a) = b, φ(b) = bb
1125761⟨a, b | aa=b, baab=abba⟩φ(a) = b, φ(b) = bb
1125763⟨a, b | aa=b, baba=aabb⟩φ(a) = b, φ(b) = bb
1125764⟨a, b | aa=b, baba=abab⟩φ(a) = b, φ(b) = bb
1125773⟨a, b | aa=b, babb=abbb⟩φ(a) = b, φ(b) = bb
1125777⟨a, b | aa=b, bbaa=aabb⟩φ(a) = b, φ(b) = bb
1125780⟨a, b | aa=b, bbab=abbb⟩φ(a) = b, φ(b) = bb
1125781⟨a, b | aa=b, bbab=babb⟩φ(a) = b, φ(b) = bb
1125782⟨a, b | aa=b, bbba=abbb⟩φ(a) = b, φ(b) = bb
1125912⟨a, b | aa=1, aaaaaaaaa=1⟩φ(a) = 1, φ(b) = b
1126440⟨a, b | aa=1, aaaaaaaa=a⟩φ(a) = 1, φ(b) = b
1126968⟨a, b | aa=1, aaaaaaa=aa⟩φ(a) = 1, φ(b) = b
1127496⟨a, b | aa=1, aaaaaa=aaa⟩φ(a) = 1, φ(b) = b
1128024⟨a, b | aa=1, aaaaa=aaaa⟩φ(a) = 1, φ(b) = b