A Lyndon’s identity theorem for one-relator monoids

Abstract For every one-relator monoid $$M = \langle A \mid u=v \rangle $$ M = ? A ? u v ? with $$u, v \in A^*$$ , ? ? we construct a contractible M -CW complex and use it to build projective resolution of the trivial module which is finitely generated in all dimensions. This proves that monoids are type $$\mathrm{FP}_{\infty }$$ FP ? , answering positively problem posed by Kobayashi 2000. We also apply our results classify cohomological dimension at most 2, describe relation module, sense Ivanov, torsion-free presentation as an explicitly given principal left ideal ring. In addition, prove topological analogues these showing satisfy finiteness property $$\mathrm{F}_\infty F classifying geometric 2. These give natural analogue Lyndon’s Identity Theorem for groups.