By Viktor A. Gorbunov
The idea of quasivarieties constitutes an autonomous path in algebra and mathematical good judgment and makes a speciality of a fragment of first-order logic-the so-called common Horn good judgment. This treatise uniformly offers the critical instructions of the speculation from a good algebraic method constructed by means of the writer himself. A progressive exposition, this influential textual content features a variety of effects by no means prior to released in publication shape, that includes in-depth remark for purposes of quasivarieties to graphs, convex geometries, and formal languages. Key beneficial properties comprise insurance of the Birkhoff-Mal'tsev challenge at the constitution of lattices of quasivarieties, important workouts, and an in depth record of references.
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B) Let us consider now the problem of characterization of semi-Chebyshev and Chebyshev (linear) subspaces G of a normed linear space E. 14 we have that the following holds. 1. , it involves also elements of E\G), it is convenient for applications, since one can deduce from it intrinsic characterizations of semi-Chebyshev subspaces in the usual concrete normed linear spaces, as we shall see below. 1, which can be found in ,  and  respectively. 1. For a closed linear subspace G of a normed linear space E the following statements are equivalent: 1°.
224-225]. 6. A linear subspace G of a normed linear space E is called a k-semi-Chebyshev subspace, respectively a k-Chebyshev subspace (where k is an integer with 0 ^ k < GO), if respectively if We recall that ^G(x) is a convex set, since it is the intersection of the two convex sets G and S(x, p(x, G}} and that for a nonvoid convex set A in a linear space E the dimension dim A is defined as the dimension of the linear subspace of E spanned by A — y, where y is an arbitrary element of A; if A = 0, then, by definition, dim A = — 1.
4. 4; see [168, pp. 213-214]) and that the converse need not hold even if dim £ < oo. Recently D. A. Ault, F. R. Deutsch, P. D. Morris and J. E. Olson  have studied best approximation by elements of interpolating subspaces, proving, among other results, the following. 20. Every interpolating subspace G of a normed linear space E is a strongly Chebyshev subspace. 19. 19 it follows that a smooth normed linear space E (in particular, the LP(T, v] spaces, for 1 < p < oo) contains no interpolating subspace.