A structural description of absorbent-continuous group-like commutative residuated lattices over complete, order-dense chains will be presented. The theorem is sharp, no further generalization is ...possible. Group-like commutative residuated lattices will be characterized as Abelian lattice-ordered groups deprived of their cancellative property only. The so-called partial-lexicographic product constructions (two of them) will be introduced, which construct group-like commutative residuated lattices. As a side-effect, it gives rise to the so-called involutive ordinal sum construction, which constructs grouplike commutative residuated lattices from a family of grouplike commutative residuated lattices. Via two decomposition theorems, corresponding to the partial-lexicographic product constructions, it will be shown that any order-dense group-like commutative residuated chain, which has only a finite number of idempotents can be built by iterating finitely many times the partial-lexicographic product constructions using solely totally ordered Abelian groups, as building blocks. The result extends the famous structural description of totally ordered Abelian groups by Hahn 4, to order-dense group-like commutative residuated chains with finitely many idempotents.
This article considers the temporal logic defined over the class of all lexicographic products of dense linear orders without endpoints and gives its complete axiomatization for it.
The definition of E-perfect effect algebras is introduced, and their structure is studied. We study the lexicographical product of an effect algebra with any upwards directed partially ordered ...Abelian group, and we show that every E-perfect effect algebra is isomorphic with such a kind of the lexicographical product.
A homogeneous factorisation of a digraph
Γ
consists of a partition
P
=
{
P
1
,
…
,
P
k
}
of the arc set
A
Γ
and two vertex-transitive subgroups
M
⩽
G
⩽
Aut
(
Γ
)
such that
M fixes each
P
i
setwise ...while
G leaves
P
invariant and permutes its parts transitively. Given two graphs
Γ
1
and
Γ
2
we consider several ways of taking a product of
Γ
1
and
Γ
2
to form a larger graph, namely the direct product, cartesian product and lexicographic product. We provide many constructions which enable us to lift homogeneous factorisations or certain arc partitions of
Γ
1
and
Γ
2
, to homogeneous factorisations of the various products.
We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding problems. The technique uses linear programs to establish separations between ...combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that this is a valid dual solution of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or coding-theoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and non-linear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and non-linear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.
Exponentiation in power series fields Kuhlmann, Franz-Viktor; Kuhlmann, Salma; Shelah, Saharon
Proceedings of the American Mathematical Society,
11/1997, Letnik:
125, Številka:
11
Journal Article
Recenzirano
Odprti dostop
We prove that for no nontrivial ordered abelian group G does the ordered power series field \R((G)) admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered ...multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
In this paper we study four properties related to the existence of a dense metrizable subspace of a generalized ordered (GO) space. Three of the properties are classical, and one is recent. We give ...new characterizations of GO-spaces that have dense metrizable subspaces, investigate which GO-spaces can embed in GO-spaces with one of the four properties, and provide examples showing the relationships between the four properties.
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