# New PDF release: A general character theory for partially ordered sets and

By K. Keimel, Karl Heinrich Hofmann

ISBN-10: 0821818228

ISBN-13: 9780821818220

We use characters of lattices (i.e. lattice morphisms into
the aspect lattice 2) and characters of topological areas
(i.e. non-stop features into an adequately topologized
element area 2) to acquire connections and dualities among
various different types of lattices and topological areas. The
objective is to provide a unified remedy of varied identified
aspects within the relation among lattices and topological areas
and to find, at the approach, a few new ones.

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Additional info for A general character theory for partially ordered sets and lattices

Sample text

If B is linearly independent, so is A. 7. Let B be any subset (finite or infinite) of V. The set B is said to be linearly independent if every finite subset of B is linearly independent. 8. (Basis of a vector space) A linearly independent set B of vectors is said to be a (Hamel) basis of V if every vector of V is a linear combination of the vectors in B. The vector space V is said to be finite dimensional if there exists a Hamel basis B consisting of finitely many vectors. It is not clear at the outset whether a vector space possesses a basis.

Uk is the same as the maximal number, h, of linearly independent vectors among the augmented vectors VI, V2,··. 6) admit a solution if and only if the maximal number of linearly independent vectors among Xl, X2,'" ,Xm is the same as the maximal number of linearly independent vectors among Xl, X2, ••• ,Xm , y. • ,Xm , Y is the same as the maximal number of linearly independent vectors among VI,V2,'" ,Vk. 6) if and only if 9 = s = h. PROOF. 6) arises in many areas of scientific pursuit. One of the pressing needs is to devise a criterion whose verification guarantees a solution to the system.

Consequently, the maximal number of linearly independent vectors among x I, X2, . , X m , Y is the same as the maximal number of linearly independent vectors among VI, V2, ... , Vk. This provides a useful characterization of consistency of a system of nonhomogeneous linear equations. 6) of equations has a solution is that the maximal number, g, of linearly independent vectors among UI, U2, • .. ,Uk is the same as the maximal number, h, of linearly independent vectors among the augmented vectors VI, V2,··.