Wong, Tak Pui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Table of Contents --- p.ii / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Definitions of Plane Tropical Curves --- p.4 / Chapter 1.1 --- Motivation --- p.4 / Chapter 1.2 --- As Varieties over the Field of Puiseux Series --- p.7 / Chapter 1.3 --- As Varieties over the Tropical Semiring --- p.10 / Chapter 1.4 --- A Combinatorial Description of Plane Tropical Curves --- p.16 / Chapter 2 --- Properties of Plane Tropical Curves --- p.21 / Chapter 2.1 --- The Degree-genus Formula --- p.21 / Chapter 2.2 --- Bezout's Theorem --- p.24 / Chapter 3 --- Non-Archimedean Amoebas and Patchworking Method --- p.28 / Chapter 3.1 --- Computing Amoebas --- p.28 / Chapter 3.2 --- Patchworking Method --- p.37 / Chapter 3.2.1 --- Maslov's Dequantization --- p.37 / Chapter 3.2.2 --- Patchworking Method --- p.39 / Chapter 4 --- Mikhalkin's Correspondence Theorem --- p.41 / Chapter 4.1 --- Parameterized Plane Tropical Curves --- p.41 / Chapter 4.2 --- Statement of the Main Theorem --- p.43 / Chapter 4.3 --- Lattice Paths --- p.46 / Chapter 4.4 --- Complex Tropical Curves --- p.51 / Chapter 4.5 --- Sketch Proof of the Main Theorem --- p.54 / Chapter 4.5.1 --- Proof of Lemma 1 --- p.55 / Chapter 4.5.2 --- Proof of Lemma 2 --- p.56 / Bibliography --- p.60

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A well known fact of classical analysis is that the Stone- Cech compacti cation of a Tychono space is achieved as the structure space of the commutative unitary ring of real valued continuous functions de ned on it. An extension of this result is obtained [1] using semiring of non-negative real valued continuous functions on the Tychono space. The present work makes a survey of this paper and attempts to extend this to the point free domain, in which it is shown that the collection of frame homomorphisms from the frame of non-negative reals to a frame is a semiring again. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

We study variants of P-frames and associated rings, which can be viewed as natural generalizations of the classical variants of P-spaces and associated rings. To be more precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either maximal or minimal. For a completely regular frame L, if the ring RL of real-valued continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented frame. These frames are less restricted than the cozero complemented frames. Using these frames we study some properties of what are called quasi m-spaces, and observe that the property of being a quasi m-space is inherited by cozero subspaces, dense z- embedded subspaces, and regular-closed subspaces among normal quasi m-space. M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal associated with I is either maximal or minimal. If all points of L are quasi P-points, we say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi P-space if and only if the frame OX is a quasi P-frame. We characterize these frames in terms of cozero elements, and, among cozero complemented frames, give a su cient condition for a frame to be a quasi P-frame. A Tychono space X is called a weak almost P-space if for every two zero-sets E and F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H. We present the pointfree version of weakly almost P-spaces. We de ne weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring RL of real-valued continuous functions on L is weakly regular. We introduce the notions of boundary frames and boundary rings, and use them to give another ring-theoretic characterization of boundary spaces. We show that X is a boundary space if and only if C(X) is a boundary ring. A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces each of which is an F-space is said to be nitely an F-space. Among normal spaces, S. Larson gave a characterization of these spaces in terms of properties of function rings C(X). By extending this notion to frames, we show that the normality restriction can actually be dropped, even in spaces, and thus we sharpen Larson's result. / Mathematics / D. Phil. (Mathematics)