共查询到20条相似文献,搜索用时 13 毫秒
1.
Michèle Giraudet 《Order》1988,5(3):275-287
Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G
k
and H
k
are isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or 1 -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available. 相似文献
2.
Manfred Droste 《Order》1988,5(3):261-273
We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H
i
which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H
i
can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S
i
) of infinite linearly ordered sets (S
i
, ) such that each group A(S
i
) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B. 相似文献
3.
It is shown that a finitely generated ordered Abelian group is generic if and only if it is superdiscrete, i.e., each homomorphic image is discretely ordered. The forcing concept uses universal sentences as forcing conditions. 相似文献
4.
We give a complete classification and construction of all normal subgroup lattices of 2-transitive automorphism groupsA(Ω) of linearly ordered sets (Ω, ≦). We also show that in each of these normal subgroup lattices, the partially ordered subset
of all those elements which are finitely generated as normal subgroups forms a lattice which is closed under even countably-infinite
intersections, and we derive several further group-theoretical consequences from our classification.
This research was supported by an award from the Minerva-Stiftung, München. The work was done during a stay of the first-named
author at The Hebrew University of Jerusalem in fall 1982. He would like to thank his colleagues in Jerusalem for their hospitality
and a wonderful time. 相似文献
5.
Jan Florek 《Algebra Universalis》2007,56(1):57-68
In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally
semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice
of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean
topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open
set in the Euclidean topology (and also the family of all order segments
, a < b, is a base for the Euclidean topology).
Received January 7, 2005; accepted in final form November 26, 2005. 相似文献
6.
The classical theorem of R. P. Dilworth asserts that a partially ordered set of width n can be partitioned into n chains. Dilworth's theorem plays a central role in the dimension theory of partially ordered sets since chain partitions can be used to provide embeddings of partially ordered sets in the Cartesian product of chains. In particular, the dimension of a partially-ordered set never exceeds its width. In this paper, we consider analogous problems in the setting of recursive combinatorics where it is required that the partially ordered set and any associated partition or embedding be described by recursive functions. We establish several theorems providing upper bounds on the recursive dimension of a partially ordered set in terms of its width. The proofs are highly combinatorial in nature and involve a detailed analysis of a 2-person game in which one person builds a partially ordered set one point at a time and the other builds the partition or embedding.This paper was prepared while the authors were supported, in part, by NSF grant ISP-80-11451. In addition, the second author received support under NSF grant MCS-80-01778 and the third author received support under NSF grant MCS-82-02172. 相似文献
7.
L. Babai 《Periodica Mathematica Hungarica》1980,11(4):257-270
With five exceptions, every finite regular permutation group occurs as the automorphism group of a digraph.One of the corollaries: given a finite groupG of ordern, there is a commutative semigroupS of order 2n+2 such that AutSG. The problem whether a latticeL of order Cn with AutLG exists (for some constantC), remains open. 相似文献
8.
Jan Florek 《Algebra Universalis》2008,58(3):341-347
On a partially ordered set G the orthogonality relation is defined by incomparability and is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space of the lattice has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We
will also prove: if G is an ordered group with a positive cone P, then the lattice has the covering property iff , where g is an element of G and M is the intersection of all maximal subgroups contained in .
Received August 1, 2006; accepted in final form May 29, 2007. 相似文献
9.
N. N. Kuzjurin 《Order》1992,9(3):205-208
I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S
1××S
n under the condition that maxi|Si|=0(n/logn). 相似文献
10.
We give a simple (and easy to apply) technique that gives the undecidability of the theory of many automorphism groups: Let
G be a group of automorphisms of a structure. Suppose that is not the identity and has no non-singleton finite orbits. If the centraliser of g is transitive on the support of g and satisfies a further technical condition, then the subgroup generated by g is equal to the double centraliser of g. Thus if G contains such an element g that is conjugate to all its positive powers, then one can interpret addition and multiplication of natural numbers in the
theory of G using the parameter g; consequently, G has undecidable theory.
Received: 9 October 2000 / in final form: 2 October 2001 / Published online: 29 April 2002 相似文献
11.
12.
Sergey S. Goncharov 《Advances in Mathematics》2003,175(1):102-143
Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or ω, that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis. 相似文献
13.
Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n – 1)
n – 1 implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n
2+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)n}, it follows thatc
1
n
2h(n)c
2
n
n+1 for somec
1 andc
2. The lower bound disproves the conjecture thath(n)2n. It is shown that if |P| – 1 spec(P) thenP has a retract of size |P| – 1 but that for all there is a bipartite ordered set with spec(P) = {|P| – 2, |P| – 4, ...} which has no proper retract of size|P| – . The case of reflexive graphs is also treated.Partially supported by a grant from the NSERC.Partially supported by a grant from the NSERC. 相似文献
14.
Norman R. Reilly 《Order》1986,3(3):287-297
It is shown that the variety
n
of lattice ordered groups defined by the identity x
n
y
n
=y
n
x
n
, where n is the product of k (not necessarily distinct primes) is contained in the (k+1)st power A
k+1 of the variety A of all Abelian lattice ordered groups. This implies, in particular, that
n
is solvable class k + 1. It is further established that any variety V of lattice ordered groups which contains no non-Abelian totally ordered groups is necessarily contained in
n
, for some positive integer n.This work was supported in part, by NSERC Grant A4044. 相似文献
15.
We show that the 2-crown is not coproductive, which is to say that the class of those bounded distributive lattices whose
Priestley spaces lack any copy of the 2-crown is not productive. We do this by first exhibiting a general construction to
handle questions of this sort. We then use a particular instance of this constrution, along with some of the combinatorial
features of projective planes, to show that the 2-crown is not coproductive.
This paper is dedicated to Walter Taylor.
Received November 24, 2004; accepted in final form July 16, 2005.
The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech
Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education
of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver. The third author would
like to express his thanks for support by the NSERC of Canada and partial support by project LN 00A056 of the Ministry of
Education of the Czech Republic. 相似文献
16.
Marius Tărnăuceanu 《Quaestiones Mathematicae》2019,42(1):87-92
In this note we give some new results concerning the subgroup commutativity degree of a finite group G. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of G. 相似文献
17.
LetX
G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(XG,H) must contain the regular subgroupL
G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(XG,H)LG] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained. 相似文献
18.
The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders. 相似文献
19.
There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille
completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for
which kind of maps the two extensions coincide.
Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that
are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies.
We then turn to expansions of lattices with additional operations, and address the question of which equational properties
of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras.
For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient
conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation
properties the primitive operations satisfy, the more equations are preserved.
Received August 21, 2005; accepted in final form October 17, 2006. 相似文献
20.
Daniele Mundici 《Journal of Pure and Applied Algebra》2007,211(2):400-403
We prove that the generators g1,…,gn of a lattice-ordered abelian group G form a free generating set iff each ?-ideal generated by any n−1 linear combinations of the gi is strictly contained in some maximal ?-ideal of G. 相似文献