Interval (mathematics) - Topological Algebra

Topological Algebra

Intervals can be associated with points of the plane and hence regions of intervals can be associated with regions of the plane. Generally, an interval in mathematics corresponds to an ordered pair (x,y) taken from the direct product R × R of real numbers with itself. Often it is assumed that y > x. For purposes of mathematical structure, this restriction is discarded, and "reversed intervals" where y − x < 0 are allowed. Then the collection of all intervals can be identified with the topological ring formed by the direct sum of R with itself where addition and multiplication are defined component-wise.

The direct sum algebra has two ideals, { : x ∈ R } and { : y ∈ R }. The identity element of this algebra is the condensed interval . If interval is not in one of the ideals, then it has multiplicative inverse . Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals in this case. The identity component of this group is quadrant I.

Every interval can be considered a symmetric interval around its midpoint. In a reconfiguration published in 1956 by M Warmus, the axis of "balanced intervals" is used along with the axis of intervals that reduce to a point. Instead of the direct sum, the ring of intervals has been identified with the split-complex number plane by M. Warmus and D. H. Lehmer through the identification

z = (x + y)/2 + j (x − y)/2.

This linear mapping of the plane, which amounts of a ring isomorphism, provides the plane with a multiplicative structure having some analogies to ordinary complex arithmetic, such as polar decomposition.