Opening and Closing

Abstract

In this memorandum we shall use the symbols ∑ and Π, precisely as in elementary algebra, to denote iterated sums and products using the operations ⊕, ⊕ ′, ⊗ and ⊗ ′. For example, if u₁, u₂,..., u_n are given (not necessarily distinct) elements of a commutative band, We shall write $\sum\limits_{i = 1}^n { \oplus ^u i}$u_i to denote the sum:

$$u_1 \oplus u_2 \oplus \ldots \oplus u_n ,$$

and $\sum\limits_{i = 1}^n { \oplus '} u_i$ u_i to denote the sum:

$$u_1 \oplus '{\text{ }}u_2 \oplus '{\text{ }}... \oplus '{\text{ }}u_n ,$$

when the dual addition is defined.