By Theorem 8.26 there is a monadic second-order relabelling ? and a transduction µ ? DTWT su such that Since T (F ) is regular, ?(T (F )) is regular by Proposition 7 This shows that L ? DTWT su (REGT) Now let L = µ(R) with µ ? By Theorem 8.6, µ is in MSOT. Since R is MS-definable, we obtain from µ by the Restriction Theorem (Theorem 7.16) an MS-transduction ? such that L = ? (T (F )). Clearly, we may assume that F is a fat signature, pp.L is VR-equational ,

(t)|} is VR-equational, see Example 8.1. Similarly, an example of a linear VRequational set of terms is {?(c n (µ A (w))), d n (µ A (w))) | w ? A * , n = |w|}, where A is any alphabet. Since deterministic tree-walking transducers with a unary output signature are single-use ,

d n are points on a circle such that, according to some orientation of the circle, d i+1 follows d i and d 1 follows d n , then C(x, y, z) expresses that, if one " walks " along the circle according to this orientation by starting at x, then one meets y before z. We denote by C(?) the ternary relation associated with ? in this way. A relation of this form is a cyclic ordering. A cyclic ordering C satisfies the following properties, for all x, y, z in its domain D, C1) C(x, y, z) ? x = y ? x = z ? y = z (C2) C(x, y, z) ? C(y ,

Its domain is D U ? T U ; for every c in R 0 , we let c S be the unique element of c U (we use here (I5)) and for every we let R S be the set of tuples) ) such that, for some t ? R U , we have (t, d i ) ? in i U for each i ? [?(R)]. It is easy to check that there is a unique isomorphism between Inc(S) and U that is the identity on D U ? T U , d ?(R) ) in T S corresponds by Conditions (I1) to (I4) to a unique element t of T U . (2) It is also easy to check that if h is an isomorphism from U to Inc(S ) for an R-structure S , then the restriction of h to D U ? T U is an isomorphism from S to S . An FO-transduction can construct S from U : Conditions (I1) to (I5) are expressible by a first-order sentence ?, that can be taken as the precondition of the definition scheme to be defined), we have given a construction of S from U in STR c (R Inc ) satisfying Conditions (I1) to (I5) We omit the details of the definition scheme. CHAPTER 9, Proof: (1) Conditions (I1) to (I5) are clearly necessary. Conversely, let us assume that they hold for some R Inc -structure U . We define as follows an Rstructure S RELATIONAL STRUCTURES Hence we can construct a (k + 1)-copying MS-transduction ? 2 that associates with (H, ?) given as V G , edg H , ?? the structure T = D T)) | (x, i) ? D T , i ? 1, y = p i (x)} ? {((x, i), (x, 0)) | (x, i) ? ,

Adj (S) ? S u be its adjacency graph Proposition 9.14) Since |E G | ? |T S |, the graph G is in US k . Thus, by Propositions 9) and 9.43 there exists a domain-preserving MS-transduction ? 1 that transforms S into one or more structures ?? satisfies the same conditions as in the above special case, with, 12 For each x ? D S , we define the enumeration p 1 (x), . . . , p l (x) of Adj ? H (x) as above. Our goal is now to define a parameterless domain-extending MS-transduction ? 2 that transforms each such structure D S , (R S ) R?R , edg H , ?? into Inc(S) ,

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