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1.
2.
Summary IfT is a complete theory of Boolean algebra, then we writeA ⊲T
B to denote that for every cardinal κ and every κ-regular filter over a setI such that the Boolean algebra 2
F
I
of all subsets ofI reduced byF is a model ofT, the reduced powerA
F
I
isK
+-saturated wheneverB
F
I
isK
+-saturated. The relation ⊲T generalizes the relation ◃ introduced by Keisler. As in the case of Keisler's ◃ it happens that ⊲T’s are relations between complete theories, i.e. ifA≡B thenA ⊲T
B andB ⊲T
A. In this paper some examples of theories which are maximal (minimal) with respect to ⊲T’s are provided and the relations ⊲T are compared with each other.
Presented by J. Mycielski 相似文献
3.
Leonid G. Fel 《Functional Analysis and Other Mathematics》2011,3(2):179-192
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r12,r1r2+r12k,r3-r12k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r
1,r
2,r
3∈ℤ+, r
1≥2 and gcd(r
1,r
2)=gcd(r
1,r
3)=1, and calculate their universal Frobenius number Φ(r
1,r
2,r
3) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe
the reduction of the minimal generating sets of these semigroups up to {r12,r3-r12k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation. 相似文献
4.
LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {T
n
}
n=1
∞
be any sequence of interval exchange transformations such thatT
1 =T andT
n
is the first return map induced byT
n-1 on one of its exchanged intervals In-1. We prove that {T
n
}
n=1
∞
contains finitely many transformations up to rescaling. If the interval In is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n
0 ∈ ℤ+, λ ∈R
+ such that for alln ≥n
0,I
n
= λI
n+k
andT
n
,T
n+k
are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms,
applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational
is eventually periodic. We also discuss the case of periodic induced sequences.
These results have applications to topology. In particular, every projective measured foliation on Thurston’s boundary to
Teichmüller space that is minimal and metrically ‘quadratic’ is fixed by a hyperbolic element of the modular group. Moreover,
if the foliation is orientable, it covers (via a branched covering) an irrational foliation of the two-torus.
We also obtain a new proof, for quadratic irrationals, of Boshernitzan’s result that a minimal rank 2 interval exchange transformation
is uniquely ergodic.
The first author was supported in part by NSF-DMS-9224667. The second author was supported in part by an NSF-NATO fellowship,
held at the Université Paris-Sud, Orsay. 相似文献
5.
We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result
is the following theorem.
Theorem: Let {Tp:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y
p
v
p
). Let S be a Cantor minimal system such that the cardinality of the set ε
S
of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ
p
:pεI} of distinct measures from ε
S
there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system,
(S
1,μ
p
) is isomorphic to Tp for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of
S are measure-theoretic factors of Tp for all p∈I.
This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families
of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T
1,T2,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T
1,T2,…}, and such that (S,μ
i
) is isomorphic toT
i for alli.
Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results
generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special
cases of our result where the collections {T
p}, {T
p
}{μ
p
} consist of just one element each.
Research of I.K. was supported by NSF grant DMS 0140068. 相似文献
6.
M. F. Gamal’ 《Journal of Mathematical Sciences》2010,165(4):435-448
In 2005, the following question was posed by Duggal, Djordjević, and Kubrusly: Assume that T is a contraction of the class C
10 such that I − T
*
T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples.
One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L
2(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem
by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T
*
T belongs to the Schatten–von Neumann classes
\mathfrakSp {\mathfrak{S}_p} for all p > 1. We give an example of a contraction T such that I − T
*
T belongs to
\mathfrakSp {\mathfrak{S}_p} for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici
and Kérchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely
continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles. 相似文献
7.
For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S
o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist
of congruences on the structure component partsI,S
o and Λ. The structure of images of this type of semigroups is also presented.
This work is supported by Natural Science Foundation of Guangdong Province 相似文献
8.
Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R.,
S
P
R,R
Q
S
,〈〉
, ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ
S
= {(s
1, s
2) ∈S×S|ss
1 = ss
2, ∀s∈S}, S' = S/ζ
S
and US-FAct = {
S
M∈S− Act |SM = M and SHom
S
(S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich
semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom
S
(S, ∐
i∈I
S) →∐
i∈I
S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism.
The research is partially supported by a UGC(HK) grant #2160092.
Project is supported by the National Natural Science Foundation of China 相似文献
9.
I. S. Ponizovskii 《Journal of Mathematical Sciences》1990,52(3):3170-3178
The semigroup algebras over a field K of the semigroups Tn of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)-1∈ K then the algebra KTn has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established,
where S is the sub-semigroup of Tn (n is arbitrary) such that S=Jn∪G where Jn={x∈Tn|rank x=1}, while G is a doubly transitive subgroup of the symmetric group Sn, the order of G being invertible in K.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR,
Vol. 160, pp. 229–238, 1987. 相似文献
10.
Mario Petrich 《Semigroup Forum》2005,71(3):337-365
On any regular semigroup S, the least group congruence σ, the greatest idempotent separating congruence μ and the least band
congruence β are used to give the T-classification of regular semigroups as follows. These congruences generate a sublattice
Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations
on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category T whose morphisms are surjective K-preserving homomorphisms subject to a mild condition. The class of regular semigroups is
made into a category T whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor
from T to T. The effect of the T-classification to P-semigroups is considered in some detail. 相似文献
11.
Leonid G. Fel 《Israel Journal of Mathematics》2011,185(1):413-444
We derive the duality relation for the Hilbert series H (d
m
; z) of the almost symmetric numerical semigroup S (d
m
) combining it with its dual H (d
m
; z
−1). We establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F
j
, generators d
i
and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately.
We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give
the necessary and sufficient conditions for the minimal set d
m
to generate such semigroups. 相似文献
12.
Ferenc Weisz 《分析论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded
from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H
1
#
(T×T), L1(T2)), where the Hardy space H
1
#
(T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H
1
#
(T×T)⊃LlogL(T
2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces
Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.
This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt
Foundation. 相似文献
13.
YangChangsen 《高校应用数学学报(英文版)》2001,16(3):285-289
Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on 相似文献
14.
A. Schrijver 《Discrete and Computational Geometry》1991,6(1):527-574
In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ2, a subset {I
1, …,I
p} of the faces ofG, and pathsC
1, …,C
k inG, with endpoints on the boundary ofI
1 ∪ … ∪I
p; find: pairwise disjoint simple pathsP
1, …,P
k inG so that, for eachi=1, …,k, P
i is homotopic toC
i in the space ℝ2\(I
1 ∪ … ∪I
p).
Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm
to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ2 and pairwise disjoint setsW
1, …,W
k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT
1, …,T
k ofG whereT
i
(i=1, …, k). 相似文献
15.
Ferenc Weisz 《逼近论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H 1 # (T×T), L1(T2)), where the Hardy space H 1 # (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H 1 # (T×T)⊃LlogL(T 2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. 相似文献
16.
L. G. Kovács 《Israel Journal of Mathematics》1988,63(1):119-127
LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT
m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT
m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally
isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic.
GivenT
m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal
subgroups are abstractly isomorphic toT
m. Secondly, form the direct productS
m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS
m×OutT whose projections inS
m are primitive. The second result of the paper is that there is a bijection between these two sets.
The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which
can fall into a single abstract isomorphism class. 相似文献
17.
We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are
definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T
ϖ} is definable, whereT
ϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite
theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable.
The research of both authors was supported by National Science Foundation Grant No. DMS-8302295 相似文献
18.
Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T
0∈B(E, F) with a generalized inverse T
0
+∈B(F, E). This paper shows that, for every T∈B(E, F) with ‖T
0
+ (T−T
0)‖<1, B ≡ (I + T
0
+(T−T
0))−1
T
0
+ is a generalized inverse of T if and only if (I−T
0
+
T
0)N(T) = N(T
0), where N(·) stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and
Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T
0 is a semi-Fredholm operator but not in general. 相似文献
19.
Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity
for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI
n, An, Dn, En and their duals for small values ofn. Although it appears thatCM(I
n)=2
n−1 ifn≤33, asn → ∞ we haveCM(I
n)∼2.089...
n
. The results have application to numerical integration. 相似文献
20.
Extensions of ordered semigroups 总被引:3,自引:0,他引:3
Adam J. Hulin 《Semigroup Forum》1971,2(1):336-342
The present note summarizes the author's dissertation [2]. Let S and T be ordered semigroups, the latter with a zero element
O. Let Σ be an ideal extension of S by T determined by a partial homomorphism ϕ of T*=T/O into S. A partial solution is given to the problem of determining all ways of extending the given orders on S and T* to Σ=S U T*. Every such “extending order” (≤) on Σ carries with it a certain “null decomposition” T*=X∪Y (X∩Y=ℴ) of T*, and the existence and behavior of extending orders is discussed in terms of properties of null decompositions of T*. 相似文献