45,941 research outputs found
Proof Theory and Ordered Groups
Ordering theorems, characterizing when partial orders of a group extend to
total orders, are used to generate hypersequent calculi for varieties of
lattice-ordered groups (l-groups). These calculi are then used to provide new
proofs of theorems arising in the theory of ordered groups. More precisely: an
analytic calculus for abelian l-groups is generated using an ordering theorem
for abelian groups; a calculus is generated for l-groups and new decidability
proofs are obtained for the equational theory of this variety and extending
finite subsets of free groups to right orders; and a calculus for representable
l-groups is generated and a new proof is obtained that free groups are
orderable
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality
We study representations of MV-algebras -- equivalently, unital
lattice-ordered abelian groups -- through the lens of Stone-Priestley duality,
using canonical extensions as an essential tool. Specifically, the theory of
canonical extensions implies that the (Stone-Priestley) dual spaces of
MV-algebras carry the structure of topological partial commutative ordered
semigroups. We use this structure to obtain two different decompositions of
such spaces, one indexed over the prime MV-spectrum, the other over the maximal
MV-spectrum. These decompositions yield sheaf representations of MV-algebras,
using a new and purely duality-theoretic result that relates certain sheaf
representations of distributive lattices to decompositions of their dual
spaces. Importantly, the proofs of the MV-algebraic representation theorems
that we obtain in this way are distinguished from the existing work on this
topic by the following features: (1) we use only basic algebraic facts about
MV-algebras; (2) we show that the two aforementioned sheaf representations are
special cases of a common result, with potential for generalizations; and (3)
we show that these results are strongly related to the structure of the
Stone-Priestley duals of MV-algebras. In addition, using our analysis of these
decompositions, we prove that MV-algebras with isomorphic underlying lattices
have homeomorphic maximal MV-spectra. This result is an MV-algebraic
generalization of a classical theorem by Kaplansky stating that two compact
Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous
[0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl
A -avoiding dimension group with an order-unit of index two
We prove that there exists a dimension group whose positive cone is not
isomorphic to the dimension monoid Dim of any lattice . The dimension
group has an order-unit, and can be taken of any cardinality greater than
or equal to . As to determining the positive cones of dimension
groups in the range of the Dim functor, the bound is optimal. This
solves negatively the problem, raised by the author in 1998, whether any
conical refinement monoid is isomorphic to the dimension monoid of some
lattice. Since has an order-unit of index two, this also solves negatively
a problem raised in 1994 by K.R. Goodearl about representability, with respect
to , of dimension groups with order-unit of index 2 by unit-regular
rings.Comment: To appear in Journal of Algebr
Positive representations of finite groups in Riesz spaces
In this paper, which is part of a study of positive representations of
locally compact groups in Banach lattices, we initiate the theory of positive
representations of finite groups in Riesz spaces. If such a representation has
only the zero subspace and possibly the space itself as invariant principal
bands, then the space is Archimedean and finite dimensional. Various notions of
irreducibility of a positive representation are introduced and, for a finite
group acting positively in a space with sufficiently many projections, these
are shown to be equal. We describe the finite dimensional positive Archimedean
representations of a finite group and establish that, up to order equivalence,
these are order direct sums, with unique multiplicities, of the order
indecomposable positive representations naturally associated with transitive
-spaces. Character theory is shown to break down for positive
representations. Induction and systems of imprimitivity are introduced in an
ordered context, where the multiplicity formulation of Frobenius reciprocity
turns out not to hold.Comment: 23 pages. To appear in International Journal of Mathematic
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