A general character theory for partially ordered sets and by K. Keimel, Karl Heinrich Hofmann

By K. Keimel, Karl Heinrich Hofmann

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

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A general character theory for partially ordered sets and lattices

We use characters of lattices (i. e. lattice morphisms into
the point lattice 2) and characters of topological areas
(i. e. non-stop capabilities into an safely topologized
element house 2) to procure connections and dualities among
various different types of lattices and topological areas. The
objective is to provide a unified remedy of varied recognized
aspects within the relation among lattices and topological areas
and to find, at the means, a few new ones.

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

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To do this, multiply termwise, Eq. (43), i. e. A. As [x', y'] and [xo, Yo] are solutions of equation (29), the result will be (x + {4y)(i - {4y) = XZ = (x' + VA y') (x' - - Ayz = {4 y') (xo + {4 Yot (xo - = (x'Z - Ay'Z)(x5 - AY5t = 1 VA Yor = (47) 31 The last step is to prove that both x and yare positive. First of all, note that x =t- 0, otherwise (47) would give us - AY5 = 1 which is impossible because A > O. Moreover, if y = 0 the same equality (47) furnishes x2 = 1, but inequalities (44) yield x > 1, a contradiction.

The factorization of the ordinary integers is of course unique. For example, 6 = 2· 3, with no other factorization being possible in the domain of ordinary integers. Consider now the set of all algebraic integers of the type 49 v=s m+n where m and n are ordinary integers, and note that both the sum and the product of two such numbers are again numbers of the same set. A set of numbers which contains any sums and products of the numbers in it is called a ring. By definition the ring under discussion contains the numbers 2, 3, 0 0.

Proof. Suppose the converse, namely, that there exists a positive integral solution [x', y'] of equation (29) such that the equality x' + VAy' = (xo + VAYo)n (38) does not hold for any positive integer n. Consider a sequence of numbers Xo + VA Yo, (xo + VA YO)2, (xo + VA Yo)3, ... It is a sequence of positive and indefinitely increasing numbers,. since xo ~ 1, Yo ~ 1 and xo + VAYo > 1. Ay')(x n - {4Yn) = x'x, - Ay'y~ + {4 (y'xn - = x'Yn) = i + {4y (43) where i and yare integers and x, - {4Yn = (xo - {4Yo)n Making use of relations (41)-(43) and inequalities (40), we obtain inequalities 1 < x + {4y < Xo + {4yo (44) We shall show that the pair of integers i and y is a solution of equation (29).

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