galois connection造句

• Every Galois connection ( or residuated mapping ) gives rise to a closure operator ( as is explained in that article ).
• In this way, closure operators and Galois connections are seen to be closely related, each specifying an instance of the other.
• Furthermore, morphisms that preserve all joins are equivalently characterized as the " lower adjoint " part of a unique Galois connection.
• A "'Galois insertion "'of into is a Galois connection in which the closure operator is the identity on.
• Thus, when one upper adjoint of a Galois connection is given, the other upper adjoint can be defined via this same property.
• In fact this approach offers additional insights both in the nature of many completeness properties and in the importance of Galois connections for order theory.
• Our considerations also yield a free construction for morphisms that do preserve meets instead of joins ( i . e . upper adjoints of Galois connections ).
• Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection.
• Another interesting way to characterize completeness properties is provided through the concept of ( monotone ) Galois connections, i . e . adjunctions between partial orders.
• A pair of adjoint functors between two partially ordered sets is called a Galois connection ( or, if it is contravariant, an " antitone"
• Another place where categorical ideas occur is the concept of a ( monotone ) Galois connection, which is just the same as a pair of adjoint functors.
• The existence of a certain Galois connection now implies the existence of the respective least or greatest elements, no matter whether the corresponding posets satisfy any completeness properties.
• The general observation on which this reformulation of completeness is based is that the construction of certain suprema or infima provides left or right adjoint parts of suitable Galois connections.
• Many antitone Galois connections arise in this way; examples include the original connection from Galois theory, the connections in linear algebra and the connection from algebraic geometry explained above.
• The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.
• Given a Galois connection with lower adjoint and upper adjoint, we can consider the compositions, known as the associated closure operator, and, known as the associated kernel operator.
• The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between and is just a monotone Galois connection between and the order dual of.
• The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between and is just a monotone Galois connection between and the order dual of.
• The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between and is just a monotone Galois connection between and the order dual of.
• A bounded lattice " H " is a Heyting algebra if and only if every mapping " f a " is the lower adjoint of a monotone Galois connection.