Complete First-Order Axiomatization of Finite or Infinite M-extended Trees
Résumé
We present in this paper an axiomatization of the structure of finite or infinite $M$-extended trees. Having a structure $M=(D_M,F_M,R_M)$, we define the structure of finite or infinite $M$-extended trees $Ext_M=(D,F,R)$ whose domain $D$ consists of trees labelled by elements of $D_M\cup F$, where $F$ is a set of function symbols containing $F_M$ and another infinite set of function symbols disjoint from $F_M$. For each $n$-ary function symbol $f\in F$, the operation $f(a_1,..,a_n)$ is evaluated in $M$ and produces an element of $D_M$ if $f\in F_M$ and all the $a_i$ are elements of $D_M$, or is a tree whose root is labelled by $f$ and whose immediate children are $a_1,..,a_n$ otherwise. The set of relations $R$ contains $R_M$ and another relation which distinguishes the elements of $D_M$ from the others. Using a first-order axiomatization $T$ of $M$, we give a first-order axiomatization $\cal{T}$ of the structure $Ext_M$ and show that if $T$ is {\em flexible} then $\cal{T}$ is {\em complete}. The flexible theories are particular theories whose function and relation symbols have some elegant properties which enable us to handle formulae more easily.
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