In the middle of the thirties, A.G. Pinsker began his scientific activities and soon he became one of the principal specialists on ordered (or semi-ordered, as they were then called) spaces. At that time, great interest was attracted in Leningrad to the study of K-spaces (Dedekind complete vector lattices) and normed lattices.

A.G. Pinsker introduced some principal notions and obtained significant results in this theory. For example, he introduced the class of extended K-spaces (universally complete vector lattices). In 1938, he proved that any K-space is embedded in some extended K-space as an order dense ideal and that this extended K-space is the maximal extension of the given K-space. A.G. Pinsker was the first to study K-spaces of countable type (order separable K-spaces). He introduced (together with H. Nakano) the notion of an order continuous (or completely linear, as it was called at that time) operator. A.G. Pinsker introduced the notions of discrete vector lattices and of continuous (atomless) vector lattices.

A.G. Pinsker payed much attention to regular K-spaces. One of the main results in this direction states that, under the Continuum Hypothesis, any regular K-space is of countable type. It seems to be the first usage of set theory hypotheses in the theory of ordered spaces. The following result was also obtained in this study: any regular Boolean algebra with a strictly positive finitely additive measure has a strictly positive countably additive measure; so this algebra is normable. This well-known result is now called Kelley's theorem, though Kelley published it only in 1959, whereas Pinsker's result may be found in the book by L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker book published in 1950. It should be noted that in this book there are some A.G. Pinsker's results obtained even before the war of 1941-1945.

It is difficult to name all results obtained by A.G. Pinsker in the theory of ordered spaces. But it should be noted that he also wrote papers on representation of ordered spaces, and on normed lattices. He presented the characterization of so called semi-ordered fields and abstract (in terms of ordered spaces) characterizations of some classes of algebras of selfadjoint operators.

The first A.G. Pinsker's paper published in 1938 contains the analytical representation of the functionals on the space of measurable functions that are additive on functions with disjoint supports (there are two papers on this subject written jointly with L.V. Kantorovich). This kind of partially additive functionals (and operators) introduced by A.G. Pinsker is studied up to now. In the theory of ordered spaces, the operators additive on disjoint elements are called disjointly additive operators. It should be noted that A.G. Pinsker payed much attention to the concept of "disjointness" in his studies. This brought him to the idea of the abstract disjointness in linear spaces. The idea to consider abstract disjointness (or orthogonality) came to algebra and analysis much later. A.G. Pinsker gave up this idea to one of his disciples (he also left some other bright ideas to his disciples).

A.G. Pinsker was also involved in the studies of l-groups. In particular, he gave the complete description of a Dedekind complete l-group (1949).

In sixties, A.G. Pinsker began to study ordered spaces of convex sets in a locally convex space. After that he was successful in the study of vector optimization, where all elements are assumed to belong to some K-space or to a more general vector lattice. His last paper in this field was published in the volume of the journal "Optimization" devoted to his memory.

A final remark. In one of the lectures of his 1954 course for graduate
students of the Leningrad University, A.G. Pinsker set up the following problem:
is it true that any extended K-space of countable type is a K_{+}-space.
Several mathematicians became interested in this problem,
but its (negative) solution was found only 28 years later.

A.I. Veksler

[2] Sur une fonctionnelle dans l'espace de Hilbert. Doklady Acad. Sci. URSS, 20 (1938), 411-414 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 20, no. 6.

[3] On an extension of a semi-ordered spaces. Doklady Akad. nauk SSSR, 21 (1938), 6-10 (in Russian).

[4] (with L.V. Kantorovich) Sur les fonctionnelles partiellement additives dans les espaces semiordonnes. C. R. A. S. (Paris), 207 (1938).

[5] (with L.V. Kantorovich) Sur les formes generales des fonctionnelles partiellement additives dans certain espaces semiordonnes. C. R. A. S. (Paris), 208 (1939).

[6] On some properties of the extended K-spaces. Doklady Akad. nauk SSSR, 22 (1939), 220-224 (in Russian).

[7] On normed K-spaces. Doklady Akad. nauk SSSR, 33 (1941), 12-15 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 33, no. 1.

[8] On a class of operations in K-spaces. Doklady Akad. nauk SSSR, 36 (1942), 243-246 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 36, 227-230.

[9] Universal K-spaces. Doklady Akad. nauk SSSR, 49 (1945), 8-11 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 49, no. 1.

[10] On the decomposition of K-spaces into elementary spaces. Doklady Akad. nauk SSSR, 49 (1945), 169-172 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 49, no. 3.

[11] On separable K-spaces. Doklady Akad. nauk SSSR, 49 (1945), 327-328 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 49, 318-319.

[12] Completely linear functionals in K-spaces. Doklady Akad. nauk SSSR, 55 (1947), 303-306 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 55, 299-302.

[13] On concrete representation of linear semiordered spaces. Doklady Akad. nauk SSSR, 55 (1947), 383-385 (in Russian). Transl.: C. R. (Doklady) Acad. Sci. URSS, 55, 379-381.

[14] On concrete representation of linear semi-ordered spaces. Uchen. zap. Pedagog. inst. im. Herzena, 64 (1948), 17-26 (in Russian).

[15] Decomposition of semi-ordered groups and spaces. Uchen. zap. Pedagog. inst. im. Herzena, 86 (1949), 235-284 (in Russian).

[16] Extensions of semi-ordered groups and spaces. Uchen. zap. Pedagog. inst. im. Herzena, 86 (1949), 285-315 (in Russian).

[17] (with L.V. Kantorovich and B.Z. Vulikh) Functional analysis in semi-ordered spaces. Moscow - Leningrad (1950), 1-548 (in Russian). Transl.: in chinese.

[18] (with L.V. Kantorovich and B.Z. Vulikh) Partially ordered groups and partially ordered linear spaces. Uspekhi mat. nauk, 6 (1951), no. 3, 31-98 (in Russian). Transl.: AMS Transl., 27 (1969), 57-124.

[19] Semi-ordered groups of the countable type. Uchen. zap. Pedagog. inst. im. Herzena, 89 (1953), 9-18 (in Russian).

[20] Regular and completely regular semi-ordered groups. Uchen. zap. Pedagog. inst. im. Herzena, 89 (1953), 19-35 (in Russian).

[21] Lattices equivalent to K-spaces. Doklady Akad. nauk SSSR, 99 (1954), 503-505 (in Russian).

[22] On conditions of equivalence of a Banach space and an L-space. Doklady Akad. nauk SSSR, 99 (1954), 677-679 (in Russian).

[23] Locally ordered groups. Trudy of third All-Union math Congress, V. 1 (1956), 32-33 (in Russian).

[24] On the representation of a K-space as a ring of selfadjoint operators. Doklady Akad. nauk SSSR, 106 (1956), 195-198 (in Russian).

[25] Lattice characterization of functional spaces. Uspekhi mat. nauk, 12 (1957), no. 1, 226-229 (in Russian). Transl.: Amer. Math. Soc. transl, 16 (1960), 442-445.

[26] (with G.P. Akilov, B.Z. Vulikh, M.K. Gavurin, V.A. Zalgaller, I.P. Natanson, D.K. Fadeev) Leonid Vitalievich Kantorovich (on the 50th anniversary of birth). Uspekhi mat. nauk, 17 (1962), no. 4, 201-209 (in Russian).

[27] On a generalization of the notion of a product for some classes of abstract spaces. Abstracts of brief sci. communications of the International math. congress, Section 5, Moscow, (1966), 68-69 (in Russian).

[28] Spaces generated by real functions on partially ordered sets. Trudy Leningrad. engen.-econom. inst., 63 (1966), 5-12 (in Russian).

[29] The space of convex sets in a locally convex space. Trudy Leningrad. engen.-econom. inst., 63 (1966), 13-17 (in Russian).

[30] The concept of a power of a K-space. Izv. VUZ, math., (1970), no. 5, 74-76 (in Russian).

[31] Boolean algebras generated by partially ordered sets. Izv. VUZ, math., (1970), no. 6, 83-85 (in Russian).

[32] The power of a metric space. Izv. VUZ, math., (1970), no. 7, 92-93 (in Russian).

[33] (with V.V. Kuzmina) Quasi-linear spaces and convex sets. Optimal planning, (1970), no. 17, 153-158 (in Russian).

[34] (with V.V. Kuzmina) Characterization of convex sets of a semiordered space. Doklady Akad. nauk SSSR, 198 (1971), 769-771 (in Russian). Transl.: Soviet Math. Dokl., 12 (1971), no. 3.

[35] (with V.V. Kuzmina) The characterization of convex sets of linear normed spaces. Izv. VUZ, math., (1972), no. 9, 90-94 (in Russian).

[36] (with B.Z. Vulikh, A.N. Kolmogorov, Ju.V. Linnik, V.L. Makarov, B.S. Mityagin, G.Sh. Rubinstein, D.K. Fadeev) Leonid Vitalievich Kantorovich (on the 60th anniversary of birth). Uspekhi mat. nauk, 27 (1972), no. 3, 221-227 (in Russian).

[37] (with V.V. Kuzmina) An intrinsic characterization of a convex set in a topological vector spaces. Optimization (1973), no. 12, 93-96 (in Russian).

[38] (with E.F. Bryzhina) Fundamental of optimal programming. Leningrad. Univers. Publ. House, (1974), 1-188 (in Russian).

[39] On the transport problem with limited transport means. Mathematics. An issue of scientific and methodical papers. M., "Higher school" (1975), issue 5, 61-63 (in Russian).

[40] Linear optimization in ordered spaces. Doklady Akad. nauk SSSR, 242 (1978), 1012-1015 (in Russian). Transl.: Soviet Math. Dokl., 19 (1978), no. 8.

[41] The transport problem in functional spaces. Sibirsk. math. Jour., 19 (1978), 1418-1420 (in Russian).

[42] (with A.I. Veksler, D.A. Vladimirov, M.K. Gavurin, L.V. Kantorovich, S.M. Lozinsky) B.Z. Vulikh. Uspekhi mat. nauk, 34 (1979), no. 4, 133-137 (in Russian).

[43] The general problem of linear programming in ordered spaces. Uspekhi mat. nauk, 34 (1979), no. 5, 219-220 (in Russian).

[44] The general problem of linear programming in ordered spaces and some of its applications. Izv. VUZ, math., (1979), no. 7, 72-75 (in Russian).

[45] A problem of linear programming with changing coefficients in the objective function. Sibirsk. mat. Zh., 20 (1979), 667-670 (in Russian).

[46] The general problem of linear optimization in K-spaces. Optimization, (1979), no. 23, 9-16 (in Russian).

[47] The specific points in presenting dynamical economical models. Leningrad. engen.-econom. Inst., (1981), 1-88 (in Russian).

[48] The dynamical transport problem. Economics and mathematical methods, 2 (1983), 363-367 (in Russian).

[49] Compact systems of problems in linear programming. Optimization, (1984), no. 34, 53-55 (in Russian).

[50] The problem of linear programming in compact metric spaces. Optimization, (1985), no. 35, 24-27 (in Russian).

[51] Linear parametrical problems of optimization in compacts. Optimization, (1986), no. 37, 58-63 (in Russian).

[53] A.I. Veksler, D.A. Vladimirov, M.K. Gavurin, L.V. Kantorovich, E.S. Lyapin, D.K. Fadeev. A.G. Pinsker. Uspekhi mat. nauk, 41 (1986), no. 2, 177-178 (in Russian).

[54] L.V. Kantorovich, G.P. Akilov, A.I. Veksler, D.A. Vladimirov, M.K. Gavurin, S.S. Kutateladze, G.Sh. Rubinstein. A.G. Pinsker's contribution to the theory of partially ordered spaces and to the vector optimization. Optimization, (1986), no. 37, 7-12 (in Russian).