Elements of Valency

In this section, we assume S to have finite valency. We shall first look at closed subsets of S which are generated by a single elements of valency 2. 

Lemma 3.3.1 Let s be a symmetric element of S, and assume that s has valency 2. Then each element of (s) has valency at most 2. 

Note finally that, by the second equation of Lemma 1.1.3(iii), 

Recall that, for each element r in (s), F[s (r ) = , s (r). Thus, Lemma 3.3.2 says, in particular, that all elements in (s) are symmetric. 

Let us now look at the structure of closed subsets of S generated by an arbitrary (not necessarily symmetric) element of valency 2. 

From Ug = 2 we obtain that (s) is 2-valenced cf. Corollary 3.1.7. Thus, assuming that 1 = 0 s)), we obtain that is odd. 

---

Valence bond theory (Section 2 3) Theory of chemical bond mg based on overlap of half filled atomic orbitals between two atoms Orbital hybridization is an important element of valence bond theory... 

Proof, (i) We are assuming that T contains a subset R of elements of valency 2 such that (R = T. 

In Section 6.5, we assume S to have finite valency. We shall prove that closed subsets of S which are generated by a single symmetric element of valency 2 are faithfully embedded in S. We also establish the corresponding recognition theorem. After that we shall look at closed subsets of S in which each nonidentity element has valency 2. 

Theorem 6.4.5 and Theorem 6.5.3 are two further recognition theorems which deal with involutions. Theorem 6.4.5 relates to George Glauberman s Z -Theorem a specific class of schemes of finite valency generated by elements of valency 2. 

A.1 NORMALIZATION CONSTANTS, ENERGIES, OVERLAPS, AND MATRIX ELEMENTS OF VALENCE BOND WAVE FUNCTIONS... 

---