Galois connections provide the order- or structure-preserving passage
between two worlds of our imagination - and thus are inherent in hu- man
thinking wherever logical or mathematical reasoning about cer- tain
hierarchical structures is involved. Order-theoretically, a Galois
connection is given simply by two opposite order-inverting (or order-
preserving) maps whose composition yields two closure operations (or one
closure and one kernel operation in the order-preserving case). Thus,
the "hierarchies" in the two opposite worlds are reversed or transported
when passing to the other world, and going forth and back becomes a
stationary process when iterated. The advantage of such an "adjoint
situation" is that information about objects and relationships in one of
the two worlds may be used to gain new information about the other
world, and vice versa. In classical Galois theory, for instance,
properties of permutation groups are used to study field extensions. Or,
in algebraic geometry, a good knowledge of polynomial rings gives
insight into the structure of curves, surfaces and other algebraic vari-
eties, and conversely. Moreover, restriction to the "Galois-closed" or
"Galois-open" objects (the fixed points of the composite maps) leads to
a precise "duality between two maximal subworlds".