Abstract
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.
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Sauer, N., Stone, M.G. Endomorphism monoids of algebras whose subalgebra lattices are chains. Algebra Universalis 28, 214–229 (1991). https://doi.org/10.1007/BF01190853
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DOI: https://doi.org/10.1007/BF01190853