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.

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

**Sample text**

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).