This thesis investigates ideals in semigroups.

Diss.--Leipzig (Andreas Thorinus, respondent). / At head of title: Q.D.B.V. Day of the month in title supplied in manuscript.

Diss.--Leipzig (Andreas Thorinus, respondent). / At head of title: Q.D.B.V. Day of the month in title supplied in manuscript.

Adademisk afhandling--Lund.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis.

Call a set of 2n + k elements Kneser colored when its n-subsets are put into classes such that disjoint n-subsets are in different classes. Kneser showed that k + 2 classes are sufficient to Kneser-color the n-subsets of a 2n + k element set. There are several proofs that this same number is necessary which rely on fixed-point theorems related to the Lusternik-Schnirelmann- Borsuk (LSB) theorem. By employing generalizations of these theorems we expand the proofs mentioned to obtain proofs of an original result we call the Subcoloring theorem. The Subcoloring theorem asserts the existence of a partition of a Kneser-colored set that halves its classes in a special way. We demonstrate both a topological proof and a combinatorial proof of this main result. We present an original corollary that extends the Subcoloring theorem by providing bounds on the size of the pieces of the asserted partition. Throughout, we formulate our results both in combinatorial and graph theoretic terminology.

A functional central limit theorem for a strictly stationary associated random field in the general d-dimension case with an added moment condition is proven. Functional central limit theorems for associated random measures are also proven. More specifically, conditions are given that imply weak convergence in the Skorohod topology of a renormalized random measure to the d-dimensional Wiener process. These results are applied to show new functional central limit theorems for doubly stochastic point random fields and Poisson cluster random measures. / Graduation date: 1986

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