l~nalshdte Kit Mh Math 99, 297-309 1985 Maihomalik 9 by Springer-Verlag 1985 Atfine Complete Semilattiees By K. Kaarli, L. Mdrki, and E. T. Schmidt, Budapest (Received 18 January 1985) Abstract: This paper presents a structural characterization of affine complete and locally affine complete semilattices. The notion of affine completeness arises naturally when investiga- ting geometric properties of (universal) algebras. Their general study was initiated by H. WERNER [5]. However, from a different starting point and without using the term "affine complete", G. GR~TZER already asked in [2], Problem 6 about characterizing affine complete algebras. Till now this problem has been solved only in some varieties with good congruence structure for the algebras (see e. g. the intro- duction of D. CLARK--H. WERNER [1] and K. KAARLI [3]). Here we present a solution for semilattices, i. e., in a variety in which congru- ences in general have no good behaviour but the operation can be easily handled. We also characterize locally affine complete semilat- tices. I. Preliminaries Throughout the paper meet semilattices will be considered. A function (of finite arity) in a semilattice S is said to be compatible if it preserves all congruences of S. Clearly, an n-ary function f in S is compatible iff (f(xl,..., xn), f(Yl,...,Yn)) ~ + O(xi,yi) for all i=1 xl,..., x,, yl,..., y, e S, where O(x, y) is the smallest congruence under which x and y are congruent (see e.g.W. N6BAUER [4]; for unary functions this condition is equivalent to saying that f preserves all 0 (x, y); the latter congruences are called the principal congruences). By a polynomial function in S we mean a function of the form a ^ x~... ^ x, where a is either the empty symbol or an element of S and the set of variables may also be empty. A local polynomial function in S is a function whose restriction to every finite subset of