Endomorphism monoids of chained graphs

Abstract For any complete chain I whose distinct elements are separated by cover pairs, and for every family {M i | i∈I} of monoids, we construct a family of graphs { G i | i∈I} such that G i is a proper induced subgraph of G j for all i,j∈I with i , G i =⋂ { G j | j∈I, i whenever i= inf { j∈I | j>i} in I , G i =⋃ { G j | j∈I, j whenever i= sup { j∈I | j in I , the endomorphism monoid of G i is isomorphic to M i for all i∈I . An analogous result is proved also for quotient graphs, and both results are applied to certain varieties of finitary algebras.

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