Endomorphism monoids of chained graphs

For any complete chain I whose distinct elements are separated by cover pairs, and for every family {Mi | i∈ I} of monoids, we construct a family of graphs {Gi | i∈ I} such that Gi is a proper induced subgraph of Gj for all i; j∈ I with i ¡ j, Gi = ⋂ {Gj | j∈ I; i ¡ j} whenever i = inf{ j∈ I | j¿ i} in I , Gi = ⋃ {Gj | j∈ I; j ¡ i} whenever i = sup{ j∈ I | j¡ i} in I , the endomorphism monoid of Gi is isomorphic to Mi for all i∈ I . An analogous result is proved also for quotient graphs, and both results are applied to certain varieties of 3nitary algebras. c © 2002 Elsevier Science B.V. All rights reserved. MSC: Primary: 05C99; Secondary: 08A35