Download e-book for kindle: Algebraic semantics by Irène Guessarian (auth.)

By Irène Guessarian (auth.)

ISBN-10: 0387102841

ISBN-13: 9780387102849

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T and t' have an upper bound, conversely, then they are compatible; if they are compatible, then they have a least upper bound and a g r e a t e s t lower bound. Proof: If and Dt, t and t' have an upper b o u n d are b o t h c o n t a i n e d in Dt,, is satisfied. b, Dt and the l a b e l i n g c o n d i t i o n if t and t' are compatible, rt(m) sup(t,t') (m) = ~t' (m) Similarly, t", then clearly of t and t' has d o m a i n DtnDt, if t(m) t(m) = t' (m) ~ inf(t,t') (m) = t otherwise. 17. M~(F,V), By its definition, T can be d e d u c e d f r o m each one of the t's in E by s u b s t i t u t i n g trees other than in t, namely, for any t in E ~ for some o c c u r r e n c e s of t~T if i n f i n i t e trees, o b t a i n e d ~ ~ is the natural e x t e n s i o n of < by saying that T~T' o b t a i n e d from T by r e p l a c i n g some o c c u r r e n c e s of trees other than ~.

Let us now apply the previous completion to the free F-magma M(F,V). 14: The c o m p l e t i o n of the F - m a g m a M ( F , V ) ordered by (cf. 1) is denoted byM~(F,V). Because of the injectivemorphismi, the order on M~(F,V) is again denoted by 4; we shall see below that it can be c h a r a c t e r i z e d in the same way as ~ on M(F,V). 12 we deduce immediately: is the free complete F - m a g m a g e n e r a t e d by V. We now investigate elements of M~(F,V). the tree like r e p r e s e n t a t i o n We need one p r e l i m i n a r y definition of and remark.

O (s,t) = t-l(s). : Let t,t' be trees on FuV and m a node in D t then denotes the tree t" o b t a i n e d by s u b s t i t u t i n g t' for the o c c u r r e n c e m in t; t" is d e f i n e d by: (ii) t(t'/v) t" (m m ' ) = t'(m') t" (m") = t(m") for e v e r y m' in D t, if m is not a left factor of m" denotes the tree t" o b t a i n e d by s u b s t i t u t i n g t' for e v e r y o c c u r r e n c e of v a r i a b l e v in t (iii) Let ~' = ' . ,tn) ' be a v e c t o r of n trees and ~=(Vl.. (tl,..

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Algebraic semantics by Irène Guessarian (auth.)


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