Isotone Analogs of Results by Mal’tsev and Rosenberg

We prove an analog of a lemma by Mal’tsev and deduce the following analog of a result of Rosenberg [11]: let Q be a finite poset with n elements, let k denote the k-element chain, and let h be an integer such that 2 ≤ h < n ≤ k. Consider the set of all order-preserving maps from Q to k whose image contains at most h elements, viewed as an n-ary relation μQ,h on k. Then an l-ary orderpreserving operation f on k preserves this relation if and only if it is either (i) essentially unary or (ii) the cardinality of f(e(Q)) is at most h for every isotone map e : Q→ k. In other words, if an increasing k-colouring of the grid k assigns more than h colours to a homomorphic image of the poset Q, then there is such an image that lies in a subgrid G1 × . . .×Gl where each Gi has size at most h, or otherwise the colouring depends only on one variable. MSC 2000: 06A11, 08A99, 08B05