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1.
Gábor Czédli 《Algebra Universalis》2009,60(1):107-124
We introduce definitions of semifractal, 0–1-fractal, quasifractal and fractal lattices. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for the other variants.
We show that a semifractal generated nondistributive lattice variety cannot be of residually finite length. This easily implies
that there are exactly continuously many lattice varieties which are not semifractal generated. On the other hand, for each
prime field F, the variety generated by all subspace lattices of vector spaces over F is shown to be fractal generated. These countably many varieties and the class of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive
fractal lattices are given each of which generates . After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals are given each of which
has cardinality and generates the variety of all lattices.
Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, thus answering
a question of Jakubík. (A convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products, a notion due to Ervin
Fried.)
This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433 and K 60148. 相似文献
2.
If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector
spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space
or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for
many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to
study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures
of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite
elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].
相似文献
3.
In Formal Concept Analysis, one associates with every context its concept lattice , and conversely, with any complete lattice L the standard context L, constituted by the join-irreducible elements as ‘objects’, the meet-irreducible elements as ‘attributes’, and the incidence
relation induced by the lattice order. We investigate the effect of the operators and on various (finite or infinite) sum and product constructions. The rules obtained confirm the ‘exponential’ behavior of and the ‘logarithmic’ behavior of with respect to cardinal operations but show a ‘linear’ behavior on ordinal sums. We use these results in order to establish
several forms of De Morgan’s law for the lattice-theoretical negation operator, associating with any complete lattice the
concept lattice of the complementary standard context.
Received February 7, 2001; accepted in final form January 6, 2006. 相似文献
4.
D. Jakubíková-Studenovská 《Algebra Universalis》2009,60(2):125-143
In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for
each element there exists a subset of such that is generated by and card .
This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by
Grant VEGA 1/3003/06. 相似文献
5.
Cancellative residuated lattices are natural generalizations of lattice-ordered
groups (
-groups).
Although cancellative monoids are defined by quasi-equations, the class
of cancellative residuated lattices is a variety.
We prove that there are only two
commutative subvarieties of
that cover the trivial variety, namely the varieties
generated by the integers and the negative integers (with zero). We also construct examples
showing that in contrast to
-groups, the lattice reducts of cancellative residuated lattices
need not be distributive. In fact we prove that every lattice can be embedded in the
lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an
order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of
.We define generalized MV-algebras and generalized BL-algebras and prove that the
cancellative integral members of these varieties are precisely the negative cones of
-groups, hence the latter form a variety, denoted by
. Furthermore we prove that the map that sends a subvariety of
-groups to the corresponding class of negative cones is a lattice
isomorphism from the lattice of subvarieties of
to the lattice of subvarieties of
.
Finally, we show how to translate equational bases between corresponding subvarieties, and
briefly discuss these results in the context of R. McKenzies characterization of categorically
equivalent varieties. 相似文献
6.
Edmond W. H. Lee 《Algebra Universalis》2009,60(2):239-246
Lyndon’s groupoid of order seven is the first published example of a non-finitely based finite algebra. The main objective
of the present article is to investigate the variety generated by this groupoid and its subvarieties. It is shown that the subvarieties of form a chain of order five, all elements of which except are Cross varieties. It follows that the variety is also generated by a groupoid of order six and that any groupoid with five or fewer elements does not generate . Consequently, Lyndon’s example of a non-finitely based finite algebra could have been of order six instead of seven. It
is also shown that, with respect to some important properties, Lyndon’s groupoid contrasts greatly with several well-known
non-finitely based finite groupoids that were discovered shortly after its publication.
Presented by R. Freese. 相似文献
7.
A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper,
we show, for a weakly closed linear subspace
of a CDCSL algebra
, that
is a Lie ideal if and only if
for all invertibles A in
, and that
is a Jordan ideal if and only if it is an associative ideal. 相似文献
8.
Let p be a prime, and let L be a set of s congruence classes modulo p. Let
be a family of subsets of [n] such that the size modulo p of each member of
is not in L, but the size modulo p of every intersection of k distinct members of
is in L. We prove that
, improving the bound due to Grolmusz and generalizing results proved for k = 2 by Snevily.
Work supported in part by the NSA under Award No. MDA904-03-1-0037. 相似文献
9.
Let be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize
the order ideals in that are well-quasi-ordered by embeddability, and thus characterize the members of that belong to at least one infinite anti-chain in .
While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was
also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839
financed by MSMT. 相似文献
10.
Given an inclusion of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of implies that of the scaling limit of . As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In
the particular case of an inclusion of local nets with the same canonical field net , we find sufficient conditions which entail the equality of the canonical field nets of .
Work supported by MIUR, GNAMPA-INDAM, the EU and SNS.
Submitted: August 29, 2008. Accepted: March 23, 2009. 相似文献
11.
Let E and F be vector lattices and
the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in
, and that if E is an AL-space and F is a Dedekind complete AM-space with an order unit, then each regular operator is a finite element in
. We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient
conditions for rank one operators to be finite elements in the vector lattice
.
A half year stay at the Technical University of Dresden was supported by China Scholarship Council. 相似文献
12.
Frédéric Naud 《Annales Henri Poincare》2009,10(3):429-451
We consider real analytic suspension semi-flows over uniformly expanding real-analytic map of the interval. We show that for any -invariant equilibrium measure related to an analytic potential g, there exists a Banach space of test functions such that for generic observables in , the corresponding correlation functions cannot decay faster than , where hg is the measure theoretic entropy of . This statement implies the existence of essential spectrum for the Perron-Frobenius operator associated to the semi-flow,
when acting on any reasonable Banach space.
Submitted: September 16, 2008. Accepted: March 30, 2009. 相似文献
13.
Lutz Strüngmann 《Archiv der Mathematik》2006,86(3):193-204
Let R be a unital associative ring and
two classes of left R-modules. In this paper we introduce the notion of a
In analogy to classical cotorsion pairs as defined by Salce [10], a pair
of subclasses
and
is called a
if it is maximal with respect to the classes
and the condition
for all
and
Basic properties of
are stated and several examples in the category of abelian groups are studied.
Received: 17 March 2005 相似文献
14.
Wolfgang Rump 《Archiv der Mathematik》2007,89(2):131-142
We introduce one-sided thick subcategories
of an arbitrary preadditive category
and define a quotient category
. When
is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick
. We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class
of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory.
Received: 14 November 2006 Revised: 12 March 2007 相似文献
15.
We define the reduced minimum modulus
of a nonzero element a in a unital C
*-algebra
by
. We prove that
. Applying this result to
and its closed two side ideal
, we get that dist
,
and
for any
if RR
= 0, where
and
is the quotient homomorphism and
. These results generalize corresponding results in Hilbert spaces. 相似文献
16.
Mohamed Bendaoud 《Archiv der Mathematik》2009,92(3):257-265
Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum
modulus of operators.
Received: 15 July 2008 相似文献
17.
Alina Iacob 《Archiv der Mathematik》2005,85(4):335-344
We consider two pairs of complete hereditary cotorsion theories
on the category of left R-modules, such that
We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules
can be computed either using a left
of M and a left
of M or using a right
a right
of N.
Received: 17 December 2004 相似文献
18.
Stefan Gille 《Archiv der Mathematik》2007,88(4):333-343
Let
be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex
then there exists a dualizing complex
of Z such that there is an isomorphism of coherent Witt groups
for all
.
Received: 3 March 2006 相似文献
19.
In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression
are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space
with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to
with n ≥ 3, where
is the scale of Hilbert spaces associated with L in
相似文献
20.
Mihran Papikian 《Archiv der Mathematik》2009,92(3):237-250
We prove a genus formula for modular curves of -elliptic sheaves. We use this formula to show that the reductions of modular curves of -elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of tends to infinity.
Received: 14 October 2008
The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship. 相似文献