In this thesis, we familiarize ourselves with lattices and Riesz spaces and prove some of their basic properties. We define Riesz homomorphisms and order ideals. Extra attention is given to order prime and maximal ideals. Our focus then shifts towards the sets of continuous functions $C(X)$ and bounded continuous functions $C_b(X)$ on a Hausdorff space $X.$ These are Riesz spaces and also real algebras, which is why their order and algebraic objects are compared. We prove that every algebra homomorphism $\varphi\colon C(X)\to C(Y)$ is a Riesz homomorphism. Moreover, for a compact Hausdorff space $K,$ we prove that every order ideal in $C(K)$ is an algebraic ideal. Additionally, we characterize closed order ideals, which in this case coincide with algebraic ones. It is shown that every algebraic prime ideal in $C(K)$ is an order prime ideal. A characterization of maximal ideals in $C(K)$ is also dealt with. Finally, we construct the Stone-Čech compactification of a Hausdorff space $X$ and use it to describe maximal ideals in the space of bounded continuous functions $C_b(X).$
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