number-theory

3B Type-substitutions 353B1 Notation As defined in 3A1 a substitution is just a finite sequence of instructions§ _ [at /al, ... , 6n/an] saying "simultaneously substitute of for at, ... , a, for an". Thefollowing extra notation will be useful.'The case n = 0 will be allowed and called the empty substitution, e. Soe(T) _ T.If n = 1, s will be called a single substitution.Each part-expression a1/al of s will be called a component of s, and called trivialif o, = ai.If all trivial components are deleted from s the resulting substitution will be calledthe nontrivial kernel of s.The set {a,,...,an} will be called Dom(s) or the variable-domain of s.And Vars(ot,...,an) will be called Range(s) or the variable-range of s.3B2 Definition Substitutions s and t are extensionally equivalent (s =ext t) if s(T)I(T) for all T.3B2.1 Lemma (i) b Dom(s) s(b) _ b.(ii) § =ext tiff s and t have the same non-trivial kernel.3B3 Definition (Restriction, s r V) If s - [at/at,...,Un/an] and V is a given setof variables, the restriction s r V of s to V is the substitution consisting of thecomponents of/ai of s such that ai E V.3B3.1 Lemma (s [ Vars(T))(T) = s(T).3B4 Definition (Union) If s _ [at/at,...,vn/an] and t _ [T,/bt,...,Tp/bp] and eithera1.... , an, bt, ... , by are all distinct or ai = b1 (Ti = r1, define(with repetitions omitted).sUt = [al/al,...,Qnlan,Tllbl,...,Tplbp]3B4.1 Lemma (i) Dom(s U t) = Dom(s) U Dom(t).(ii) If § =ext s' and I =ext t' and s U t is defined, then so is s' U C and§ U t =ext § U t'.The next definition will be the composition of two substitutions s and t, asimultaneous substitution that will have the same effect as applying t and s insuccession. To motivate it, consider the case§ - [(c-+d)/a, (b->a)/b], I - [(b->a)/b]and let T = a-+b. The result of applying first t then s is easily seen to bes(t(T)) _ (c->d)-(b--*a)-+(c->d);There seems to be no standard substitution notation in the literature.