We study the relationship between l-groups, Bézout domains, and MV -algebras. Our main motivation and starting point has been the Jaffard-Ohm correspondence between Abelian l-groups and Bézout domains ¶ μ 3 ́ Bézout domains with non-zero unit radical ¶ μ 3 ́ Unital l-groups ¶ μ 3 ́ MV -algebras ©© ©© ©© © HHHHHH ——————————– and Mundici’s equivalence between Abelian unital l-groups and MV -algebras...

In the late 1960’s, Robert Langlands introduced a number of ideas to the theory of automorphic forms and formulated a number of conjectures which gave the theory a new focus. I was a colleague of his at this time, and a good deal of my professional energy since then has been directed to problems posed by him. Thus it was not entirely inappropriate that when I was invited to this conference, Miy...

This paper presents symmetry classes of the Hartree-Fock (HF) solutions of spin and orbital ordered states in a t2g Hubbard model on a two-dimensional square lattice. Using a group theoretical bifurcation theory of the Hartree-Fock equation, we obtained many types of broken symmetry solutions which bifurcate from the normal state through one step transition in cases of commensurate ordering vec...

An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G Γ such that Γ is an Archim...

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore,...

A partially ordered group G = (G, +, ≤) is both a (not necessarily Abelian) group (G, +) (with binary operation + and identity element 0, where the inverse of a member a of G is denoted by −a) and a partially ordered set (G, ≤) in which a ≥ b implies that a+ c ≥ b+ c and c+ a ≥ c+ b for a, b, and c in G. An element a of G is positive if a ≥ 0. A partially ordered group (G, +, ≤) is called a lat...

We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a nonsurjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through t...

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be `-embedded in an `-simple lattice-or...

A unital l-group (G,u) is an abelian group G equipped with a translation-invariant lattice-order and a distinguished element u, called orderunit, whose positive integer multiples eventually dominate each element of G. We classify finitely generated unital l-groups by sequences W = (W0,W1, . . .) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Al...