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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**Example text**

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 [46], [163] and [81] 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 [5] 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.