AA* Relations

Abstract

Let E_l be a self-conjugate belt. We shall say that E_l is pre-residuated if it satisfies the following axiom X₁₃:

X₁₃: If J ⊂ E₁ is a finite set of elements, then:

$$\left. \begin{matrix} Vx\varepsilon {{E}_{1}},\left[ \underset{a\varepsilon J}{\mathop{{{\Sigma }_{\oplus '}}}}\,\left( {{a}^{*}}\otimes 'a \right) \right]\otimes x\ge x\\ x\otimes \left[ \underset{a\varepsilon J}{\mathop{{{\Sigma }_{\oplus '}}}}\,\left( a\otimes '{{a}^{*}} \right)\right]\ge x\\ \end{matrix} \right\}$$

(8-1)

The reason for choosing the term pre-residuated will emerge in Chapter 9.