1 galois connections for operations and relations r. Pöschel abstract this paper reports on various galois connections between operations and. Galois connections examples properties gc and abstract interpretation monotone galois connection let hx. Galois connection if, for all x 2x. L is called the lower adjoint. And u is called the upper adjoint.
Four levels of galois connections are exhibited, starting with the classical one and going via concrete galois connections to galois adjunctions. A galois connection is rather weak compared to an order isomorphism between the involved posets, but every galois connection gives rise to an isomorphism of certain sub. Posets, as will be explained below. The literature contains two closely related notions of.
In order theory the term galois connection. Due to ore 44, who spelled it. Adjunction between posets. Dual adjunction between posets. The former notion is sometimes called. Monotone galois connection. Antitone galois connection. Galois connections are ubiquitous. Together with adjunctions, their close rela. Tives, occur in a number of research areas, ranging from the most theoretical to the most applied.
Finally the galois connexion is perfect when it is perfect in both p and q. It is an important problem in the application of the theory of galois con. Nexions to determine when a given galois connexion is perfect. Stance represents the main content of the ordinary galois theory of equations. We start by recalling the classical descriptions of galois connections by g. Ore in their covariant forms. I galois connections of the first kind.