The paper deals with several mathematical problems associated with a mathematical model of data in computing systems (nominative sets) used in the composition-nominative approach to software system formalization. In this paper the structure of the partially-ordered set of nominative sets is investigated in terms of set theory, lattice theory and algebraic systems theory. To achieve this aim the correct transferring of basic set-theoretic operations to nominative sets is investigated in detail. A basic partially ordered algebraic system intended to provide correct analogues of basic set-theoretic operations for nominative sets is elaborated and its structure is investigated. It is established that this structure is a lower semilattice. Properties of lower and upper cones of subsets of the poset of nominative sets are investigated in detail. Subsets for which upper cones are non-empty are characterized. Closed intervals of nominative sets are examined. Boolean algebra is determined on any such interval. A criterion for isomorphism of two closed intervals is obtained. Maximal closed intervals are investigated. It is established that the poset of nominative sets is a union of isomorphic overlapping maximal closed intervals.