The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not o...

A large number of papers have been published that are devoted to showing that certain algebraic objects obtained by defining operations on the set of all continuous real-valued functions on a suitably restricted topological space determine the space. We mention but a few of them below. Let C(X), L(X), S(X) denote respectively the ring, lattice, multiplicative semigroup of all continuous real-va...

Abstract. In this paper we study the set of functions G-valued which can be approximated by G-valued continuous functions in the norm L∞G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain G-valued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem. Also, we characterize the set of functions whic...

Abstract Given a commutative semiring with compatible preorder satisfying version of the Archimedean property, asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together map from to ring continuous functions. Strassen’s theorem characterizes relaxation that asymptotically compares large powers elements up subexponential...

Pseudorandom encoding with quantized real modulation values encodes only continuous real-valued functions. However, an arbitrary complex value can be represented if the desired value is first mapped to the closest real value realized by use of pseudorandom encoding. Examples of encoding real- and complex-valued functions illustrate performance improvements over conventional minimum distance map...

for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xER there is an xaER such that xx°x = x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring R, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between th...