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October 24, 2016 / porton

A new little theorem (Galois connections)

I’ve added the following to my research book:

Definition
Galois surjection is the special case of Galois connection such that f^{\ast} \circ f_{\ast} is identity.

Proposition
For Galois surjection \mathfrak{A} \rightarrow \mathfrak{B} such that \mathfrak{A} is a join-semilattice we have (for every y \in \mathfrak{B})

f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x = y \}.

(Don’t confuse this my little theorem with the well-known theorem with similar formula formula f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x \leq y \}.)

This formula in particular applies to the Galois connection between funcoids and reloids (see my book).

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