An ideal of N -tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N -tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class CDD N of all unitary equivalence classes of such N -tuples are established and certain ideals in CDD N are distinguished. It is proved that infinite operations in CDD N may be reconstructed from the direct sum operation of a pair. Prime decomposition in CDD N is proposed and its (in a sense) uniqueness is established. The issue of classification of ideals in CDD N (up to isomorphism) is discussed. A model for CDD N is described and its concrete realization is presented. A new partial order of N -tuples of operators is introduced and its fundamental properties are established. Extremal importance of unitary disjointness of N -tuples and the way how it 'tidies up' the structure of CDD N are emphasized. 2010 MSC: Primary 47B99; Secondary 46A10. Key words: closed operator; densely defined operator; unitary equivalence; direct sum of operators; direct integral; decomposition of an operator; prime decomposition of an operator; finite system of operators.

Abstract. The aim of the paper is to prove that the bounded and unbounded Urysohn universal spaces have unique (up to isometric isomorphism) structures of metric groups of exponent 2. An algebraic-geometric characterization of Boolean Urysohn spaces (i.e. metric groups of exponent 2 which are metrically Urysohn spaces) is given.

The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G_r(N), is investigated and it is shown that each of G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are studied and it is proved that every separable metrizable topological vector space may be enlarged to G_r(0) with a `linear-like' structure which extends the linear structure of the given space.Comment: 60 page