TT PRELIMINARIES We suppose the ordinal numbers to be defined so that each ordinal £ is the set of all of its predecessors, i.e., £ = {rj : rj £}. In particular, n = {0,..., n 1} for every natural number n. The set of all natural numbers will be denoted by u). The greatest integer less than or equal to a given real number x is denoted, as usual, by [x\ and the least integer greater than or equal to x by \x\. For each set X, we write |X| for the cardinality of X, and Sb (X) for the collection of all subsets of X. Given sets X and F , we denote their set-theoretic difference by X ~ Y, i.e., X~Y = {x G X : x £ Y} when X is a fixed universe of discourse, we write "~Y" instead of "X~Y". We denote the Cartesian product of X and Y by I x F , i.e., X xY = {(#, y) : x G X and y G Y}. The set of functions from X to Y is denoted by X Y. If / is a function whose domain X and range Y include U and V respectively, then f[U] is the image of U under / , i.e., {f(u) : w G 17}, and / _ 1 [V] is the inverse image of V under / , i.e., {u G X : /(u) G F } . If a (a$ : £ «) and b = (brj : rj X) are two (ordinally indexed) sequences, then a"6 denotes their concatenation, i.e., the sequence c with domain A C + A such that c^ = a^ when £ /c, and e^ = 6^ when £ = K + r and rj X. We shall identify binary relations on a set U with subsets of 17 x [7, and we shall usually use the letters P, 2, JR, 5, T, possibly subscripted, to denote binary relations. The relative product or relational composition of two binary relations R and 5, and the converse, or inverse of such a relation are defined in the usual way: R\S = {(#, 2/) : there is a z such that (x, z) e R and (z, y) G S}, /J _ 1 = {(a:,y):(2/,a )€iJ}. The identity and diversity relations on 17 will be denoted by Iv and Dv respectively, or simply by I and D, when no confusion can arise. Notice that, for functions F and G from 17 to 17, the functional composition FoG is not the same as the relational composition F\G\ in fact, F|£? = GoF. Capital German letters will denote algebras and the corresponding Roman let- ters their universes. For example, A is the universe of 51. Occasionally, following standard practice, we shall refer to A as if it were the algebra. For instance, when dealing with a group (3, we may talk about a subset / / of G being a subgroup. An algebra is said to be trivial if its universe has only one element and non-trivial otherwise. The reduct of an algebra 21 = (A, O ^ e s to a subset T of its operations (more precisely, to a subset T of its index set) is the algebra 21' = (A, 0^)^er- Given a class K of algebras, the class of all algebras isomorphic to homomorphic images, subalgebras, or direct products of systems of members of K is denoted by H(K), S(K), or P(K) respectively. An algebra is n-generated if it is generated by l
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