Residuation Theory

Residuation Theory

Volume 102 in International Series of Monographs on Pure and Applied Mathematics
1972, Pages 211-360
Residuation Theory

CHAPTER 3 - RESIDUATED ALGEBRAIC STRUCTURES

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This chapter presents the residuated algebraic structures. An ordered groupoid mean a groupoid G that is also an ordered set in which, for each x ∈ G, the translations are isotone mappings. An ordered groupoid G is residuated on the left if each left translation on G is a residuated mapping and residuated, if it is residuated on both the left and right. Every Boolean algebra is residuated with respect to . It is readily verified that for all x, z the set {y; x ∩ y ≤ z} is not empty and admits a maximum element, namely z: x = z ∩ x′. If Rf denotes the equivalence relation associated with f then Rf is a closure equivalence on E and can be characterized as an equivalence relation with convex classes such that each class modulo Rf contains one and only one element of Im f+ that is the greatest element in its class.

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