Endomorphism monoids of algebras whose subalgebra lattices are chains

A monoidM and a latticeL arealgebraic if there is an algebraA with endomorphism monoid EndA ? M and subalgebra lattice SuA ?L. For each chainC we characterize those monoidsM for whichM and C are algebraic. In particular we show that a finite monoidM is algebraic with the three-chain iff the “equalizers” ofM form a chainE =3. The same assertion however fails for infinite monoids. This generalizes the corresponding result for two-chains and solves a problem posed by B. Jónsson ([2], p. 147). We settle the same question for all longer chainsK.