Hahn embedding theorem

Abstract: Hahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented.

Abstract: The name of Hans Hahn (1879–1934), an Austrian mathematician, a Professor of Chernivtsi (1909–1916), Bonn (1916–1921) and Vienna (1921–1934) Universities is well known among mathematicians mainly due to the famous Hahn-Banach Theorem on extensions of linear functionals. Much less known is the fact that H. Hahn independently of S. Banach proved another basic principle of Functional Analysis the uniform boundedness principle. Some other well-known results due to H. Hahn are: the Hahn decomposition theorem, the Vitali-Hahn-Saks theorem in Measure Theory, the Hahn-Mazurkiewicz theorem on continuous images of the unit segment in Topology, the Hahn embedding theorem in the Theory of Partially Ordered Sets. The notions of local connectivity and reflexivity introduced by Hahn also play an important role in modern mathematics. H. Hahn was a very versatile mathematician. His scientific heritage contains papers in Calculus of Variations, Real Functions Theory, Functional Analysis, Topology, History and Philosophy of Mathematics. In honour of the memory of Hans Hahn, mathematicians from Chernivtsi National University (Ukraine) organized regular conferences, beginning in 1984. The first and the second conferences dedicated to the memory of H. Hahn were held in Chernivtsi in 1984 and 1994, respectively.

Abstract: Let be the class of odd involutive even the notion of partial lex products is not sufficiently general. One more tweak is needed, a slightly even more complex construction, called partial sublex product, introduced here.