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Galois Connections and Pair Algebras

Published online by Cambridge University Press:  20 November 2018

J. C. Derderian
J. C. Derderian*
Affiliation:
State University of New York at Buffalo, Buffalo, New York
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Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” .

An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that

(RM 1) xϕψ ≧ x for all x i n P;

(RM 2) yψϕ ≦ for all y in Q.

Let Q* denote the partially ordered set with order relation dual to that of Q.

(A) The following conditions are equivalent:

(i) ϕ: P → Q* is a Galois connection;

(ii) ϕ: P → Q is a residuated mapping;

(iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ.

Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 498 - 501
Copyright
Copyright © Canadian Mathematical Society 1969

References

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