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.

Highlights

  • Hahn’s celebrated embedding theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup into the Hahn product → ×H Ω R, where R is the additive group of real numbers, Ω is the set of archimedean equivalence classes of G, → ×H Ω R is the set of all functions from Ω to R which vanish outside a well-ordered set, endowed with a lexicographical order [12]

  • The generalization will be a by-product of a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of the partial lexicographic product construction and using only linearly ordered abelian groups

  • Prominent examples of odd involutive FLe-algebras are odd Sugihara monoids and lattice-ordered abelian groups. The latter constitutes an algebraic semantics of Abelian Logic [4,19,21] while the former constitutes an algebraic semantics of IUML∗, which is a logic at the intersection of relevance logic and many-valued logic [8]

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Summary

Introduction

Hahn’s celebrated embedding theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup into the Hahn product → ×H Ω R, where R is the additive group of real numbers (with its standard order), Ω is the set of archimedean equivalence classes of G (ordered naturally by the dominance relation), → ×H Ω R is the set of all functions from Ω to R (alternatively the set of all vectors with real elements and with coordinates taken from Ω) which vanish outside a well-ordered set, endowed with a lexicographical order [12]. In odd involutive FLe-algebras x → x is somewhat reminiscent to a reflection operation across a point in geometry, yielding a symmetry of order two with a single fixed point In this sense t = f means that the position of the two constants is symmetric in the lattice. The latter constitutes an algebraic semantics of Abelian Logic [4,19,21] while the former constitutes an algebraic semantics of IUML∗, which is a logic at the intersection of relevance logic and many-valued logic [8] These two examples represent two extreme situations from another viewpoint: There is a single idempotent element in any lattice-ordered abelian group, whereas all elements are idempotent in any odd Sugihara monoid. For this class a representation theorem along with some corollaries (e.g. the generalization of Hahn’s theorem) will be presented in this paper

Odd Involutive FLe-algebras Versus Partially Ordered Abelian Groups
Two Illustrative Examples
Constructing Involutive FLe-algebras—Partial Lexicographic Products
Subuniverses
Extremals
Gaps—Motto: “Not All Gaps Created Equal”
One-Step Decomposition—when u is Idempotent
10. One-Step Decomposition—When u is Not Idempotent
11. Group Representation
Full Text
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