Existence of digital extensions of semi-modular state charts
Mitsuhiro Hattori · Journal of the Mathematical Society of Japan · 1971
provide the proof of Theorem 1.2 at the end of this section, where the more detailed form Theorem 1.2' will be given.DEFINITION 1.3.Let (V, h) and $(D, h)$ be semi-modular state charts such that (V, $h$ ) $\subset(D, h)$ and let $(D^{k}, h^{k})$ be an extension of $(D, h)$ .We define a subset $V^{k}$ of $D^{k}$ by $V^{k}=\{A^{k}|A^{k}\in D^{k}, A^{k}|J\in V\}$ .$V^{k}$ will be written as $\lambda_{V}(D^{k})$ or simply $\lambda(D^{k})$ when there is no confusion.We say $V^{k}$ the induced extension of $V$ by $D^{k}$ , or $V^{k}$ is induced by $D^{k}$ .Furthermore, let $\lambda_{V}(h^{k})=h^{k}|\lambda_{V}(D^{k})$ .By the lemma below, if $D^{k}$ is distributive then $\lambda_{V}(D^{k})$ is semi-modular andis called the induced extension of (V, h) by $(D^{k}, h^{k})$ .As we forementioned at the beginning of this section, we often writeand$D^{k}$ are distributive.PROOF.It is clear that $0\prime l^{k}\in V^{k}$ .Moreover, if $M^{k}$ and $N^{k}$ are states of $V^{k}$ , then $M^{k}\vee N^{k}\in D^{k}$ and $(M^{k}\vee N^{k})|J=(M^{k}|J)\vee(N^{k}|J)\in V$ and thus $M^{k}\vee N^{k}$ $\in V^{k}$ .Therefore, to prove that $V^{k}$ is semi-modular, it remains to show that $N^{k}=M^{k}+\delta^{p}$ for some $p\in J^{k}$ if $N^{k}$ covers $M^{k}$ in $V^{k}$ .Let us consider in $D^{k}$ a covering sequence $M^{k}=C(0)^{k},$ $C(1)^{k},$ $\cdots$ , $C(r)^{k}=N^{k}$ .If $N^{k}|J=M^{k}|J$ then $C(1)^{k}|J=M^{k}|J\in V$ and hence $C(1)^{k}\in V^{k}$ .This implies, however, $N^{k}=M^{k}+\delta^{p}$ for some $p\in J^{k}$ .Suppose that $M^{k}|J 1$ , then $[\theta-1, i]$ is in $\sigma(V)$ and $[\theta-1, i]<[\theta, i]$ (see 7.4 and 7.5 of [6]).Assume that there are infinite number of $[\theta, i]Thus from the definition of $\sigma(V),$ $[\varphi, j]$ cannot be in $\sigma(V)$ , contradicting the assumption.Therefore the number of $[\theta, i]\in\sigma(V)$ such that $[\theta, i]\leqq[\varphi, j]$ must be finite for every $[\varphi, j]$ of $\sigma(V)$ , and then $\sigma(V)$ is a change diagram.LEMMA 1.8.Let $V$ be a semi-modular subset of $W^{J}$ and let $D$ be a (tis- tributive subset of $W^{J}$ such that $D\supset V$ .If $[\theta, i]$ and $[\varphi, j]$ are changes of $\sigma(V)$ and $[\theta, i]\leqq[\varphi, j]$ in $\sigma(D)$ , then $[\theta, i]\leqq[\varphi, j]$ in $\sigma(V)$ .