ABSTRACT
In Section 8.5, the last section of the preceding chapter, we introduced the notions of left and right ordered representations of a system of relations. Let us briefly summarize the definitions and results given there. For a given system Σ of relations,
operators Zi,. . . ,Zn (respectively, r b . . . , r n) acting in the space T n of n-ary symbols are called operators of left (respectively, right) ordered representation if
1 n for any n-ary symbol f ( x χ, . . . , x n) Є T n whenever A — (Αχ, . . . , A n) is an n-tuple of T - generators satisfying Σ. If a relation system Σ admits a left (or right) ordered representation by ^■-generators in T n, then the set
is an algebra, and the composition law in this algebra is expressed in terms of symbols by the formula
. In most cases, one
in particular, this is always the case if the Jacobi condition holds. Thus, the existence of an ordered representation permits one to “commute” the arguments of /
(or g) in the product
1 n to their respective places so as to obtain a single expression of the form h(A \ , . . . , A n). For that reason, relation systems admitting an ordered representation (left, or, equivalently, right) will be called systems of commutation relations, or, for short, simply commutation relations.
