Equational bases and nonmodular lattice varieties
Ralph
McKenzie
1-43
Abstract: This paper is focused on equational theories and equationally defined varieties of lattices which are not assumed to be modular. It contains both an elementary introduction to the subject and a survey of open problems and recent work. The concept of a ``splitting'' of the lattice of lattice theories is defined here for the first time in print. These splittings are shown to correspond bi-uniquely with certain finite lattices, called ``splitting lattices". The problems of recognizing whether a given finite lattice is a splitting lattice, whether it can be embedded into a free lattice, and whether a given interval in a free lattice is atomic are shown to be closely related and algorithmically solvable. Finitely generated projective lattices are characterized as being those finitely generated lattices that can be embedded into a free lattice.
Some remarks concerning the varieties generated by the diamond and the pentagon
S. D.
Comer;
D. X.
Hong
45-54
Abstract: In 1945 M. P. Schützenberger exhibited two identities. He asserted that one provided an equational base for the diamond $ {M_3}$ and the other a base for the pentagon ${N_5}$. Recently Ralph McKenzie produced another equational base for ${N_5}$. In the present paper the authors modify McKenzie's idea to verify Schützenberger's assertion for ${M_3}$. They also show Schützenberger's claim about ${N_5}$ is false.
Topologies of closed subsets
Louis
Narens
55-76
Abstract: In this paper various topologies on closed subsets of a topological space are considered. The interrelationships between these topologies are explored, and several applications are given. The methods of proof as well as some intrinsic definitions assume a familiarity with A. Robinson's nonstandard analysis. E. Michael (Topologies of spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182), K. Kuratowski (Topology, Vols. I and II, Academic Press, New York, 1968), L. Vietoris (Berichezweiter Ordnung, Monatsh. Math.-Phys. 33 (1923), 49-62), and others have considered methods of putting topologies on closed subsets of a topological space. These topologies have the property that if the underlying topological space is compact then the topology of closed subsets is also compact. In general, however, these topologies of closed subsets are not compact. In this paper, a topology of closed subsets of a topological space is constructed that is always compact. This topology is called the compact topology and has many pleasant features. For closed subsets of compact Hausdorff spaces, this topology agrees with Vietoris' topology. For arbitrary spaces, there are interesting connections between the compact topology and topological convergence of subsets, including generalized versions of the Bolzano-Weierstrass theorem.
Real length functions in groups
Nancy
Harrison
77-106
Abstract: This paper is a study of the structure of a group G equipped with a 'length' function from G to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon's work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $\vert g\vert$ such that the following axioms are satisfied: \begin{displaymath}\begin{array}{*{20}{c}} {{{\text{A}}_0}.\quad \vert x\vert < ... ... \geq m\;{\text{imply}}\;c(x,z) \geq m.} \end{array} \end{displaymath} In this paper structure theorems are obtained for the cases when G is abelian and when G can be generated by two elements. We first prove that if G is abelian, then G is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of G. We prove that if G can be generated by two elements, then G is either free or abelian.
Smooth embeddings of homologically similar manifolds
Dennis M.
Roseman
107-126
Abstract: We consider the situation where we have two smooth n-manifolds $N \subseteq M$ with ${H_\ast }(M,N) = 0$ and show that given a smooth embedding of N into some manifold Q we may, under suitable conditions, extend this to embeddings of M into Q, $Q \times I$, or $Q \times {I^2}$ (where I is the unit interval). We can apply these results to obtain smooth embeddings of homologically k-connected manifolds into $(2n - k + 1)$-dimensional euclidian space.
Normed convex processes
Stephen M.
Robinson
127-140
Abstract: We show that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.
Quasi-complemented algebras
T.
Husain;
Pak-Ken
Wong
141-154
Abstract: In this paper we introduce a class of algebras which we call quasi-complemented algebras. A structure and representation theory is developed. We also study the uniformly continuous quasi-complementors on ${B^\ast}$-algebras.
Cones and Vietoris-Begle type theorems
D. G.
Bourgin
155-183
Abstract: Infinite cone constructions are exploited to yield diverse generalizations of the Vietoris-Begle theorem for paracompact spaces and Abelian group sheaves. The constructions suggest natural space, map classifications designated as almost p-solid. The methods are extended to upper semicontinuous closed multivalued maps and homotopies and culminate in a disk fixed point theorem for possibly nonacyclic point images.
The Lefschetz fixed point theorem for noncompact locally connected spaces
R. J.
Knill
185-198
Abstract: Leray's notion of convexoid space is localized and used to show that if $ f:M \to M$ is a relatively compact map on a locally convex manifold M, and f has no fixed points then its Lefschetz trace is zero. A similar theorem holds for certain adjunction spaces $ Y{ \cup _g}Z$ where Y is Q-simplicial and Z is locally convexoid. A number of other properties of locally convexoid spaces are derived; for example, any neighborhood retract of a locally convexoid space is locally convexoid.
Near central automorphisms of abelian torsion groups
Jutta
Hausen
199-215
Abstract: This paper is concerned with the normal structure of the automorphism group $A(T)$ of an abelian torsion group T. The concept of the near center of a group is introduced in order to determine all subgroups of $A(T)$ the centralizer of which has finite index. Consequences are the fact that the finite normal subgroups of $A(T)$ are nilpotent if T is a primary group of infinite rank, and that every normal torsion subgroup of $A(T)$ is contained in the center of $A(T)$ if T is divisible.
Instability in ${\rm Diff}\sp{r}$ $(T\sp{3})$ and the nongenericity of rational zeta functions
Carl P.
Simon
217-242
Abstract: In the search for an easily-classified Baire set of diffeomorphisms, all the studied classes have had the property that all maps close enough to any diffeomorphism in the class have the same number of periodic points of each period. The author constructs an open subset U of $ {\text{Diff}^r}({T^3})$ with the property that if f is in U there is a g arbitrarily close to f and an integer n such that ${f^n}$ and ${g^n}$ have a different number of fixed points. Then, using the open set U, he illustrates that having a rational zeta function is not a generic property for diffeomorphisms and that $\Omega $-conjugacy is an ineffective means for classifying any Baire set of diffeomorphisms.
Singular homology as a derived functor
G. S.
Rinehart
243-256
Abstract: A general theory of relative derived tunctors is applied to the category of topological spaces to obtain singular homology and cohomology, verify the Eilenberg-Streenrod axioms, and show that singular and simplicial theory agree.
Semi-$p$-functions
J. F. C.
Kingman
257-273
Abstract: A generalisation of the theory of p-functions which applies, for instance, to the diagonal elements of one-parameter semigroups of infinite matrices which satisfy no boundedness condition.
Algebras of analytic germs
William R.
Zame
275-288
Abstract: Let S be a Stein-Riemann domain with global local coordinates $ {\sigma _1}, \cdots ,{\sigma _n}$. Let X be a compact subset of S. Denote by $ \mathcal{O}(X)$ the algebra of germs on X of functions analytic near X. A subalgebra of $ \mathcal{O}(X)$ containing the germs of ${\sigma _1}, \cdots ,{\sigma _n}$ and the constants is stable if it is closed under differen tiation with respect to the coordinates ${\sigma _1}, \cdots ,{\sigma _n}$. In this paper the relationship of a stable algebra to its spectrum is investigated. In general, there is no natural imbedding of the spectrum into a Stein manifold. We give necessary and sufficient conditions that such an imbedding exists, and show that a stable algebra whose spectrum admits such an imbedding has a simple description. More generally, we show that a stable algebra is determined by its spectrum. This leads to certain approximation theorems.
A pair of indices for function spaces on the circle
Colin
Bennett
289-304
Abstract: We give here some of the basic properties of the classes $\{ {\Phi _r}\}$, $\{ {\Psi _r}\} , - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces $\mathfrak{X}$ on the circle It is shown that to each space $ \mathfrak{X}$ there correspond two numbers $\xi ,\eta$, called indices, which satisfy $0 \leq \eta \leq \xi \leq 1$; these numbers represent the rate of growth or decay of $\left\Vert {{\Psi _r}} \right\Vert$ as $r \to \pm 1$. By using the operators $ {\Psi _r}$ to obtain estimates for certain averaging operators ${A_\gamma }$, we are able to show that the indices $(\xi ,\eta )$ coincide with the Boyd indices $(\alpha ,\beta )$. As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.
Constructive proof of Hilbert's theorem on ascending chains
A.
Seidenberg
305-312
Abstract: In a previous note it was shown that if a bound $f(i)$ is placed on the degrees of the elements in some basis of an ideal ${A_i}$ in the polynomial ring $k[{X_1}, \cdots ,{X_n}]$ over an explicitly given field $k,i = 0,1,2, \cdots$, then a bound can be (and was) constructed for the length of a strictly ascending chain ${A_0} < {A_1} < \cdots$. This result is now obtained using a strictly finitist argument. A corollary is a finitist version of Hilbert's theorem on ascending chains.
Strictly regular elements in Freudenthal triple systems
J. C.
Ferrar
313-331
Abstract: Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra ${E_7}$) over finite, p-adic or real fields.
The $L\sp{p}$ behavior of eigenfunction expansions
Harold E.
Benzinger
333-344
Abstract: We investigate the extent to which the eigenfunction expansions arising from a large class of two-point boundary value problems behave like Fourier series expansions in the norm of $ {L^p}(0,1),1 < p < \infty$. We obtain our results by relating Green's function to the Hilbert transform.
Adequate ultrafilters of special Boolean algebras
S.
Negrepontis
345-367
Abstract: In his paper Good ideals in fields of sets Keisler proved, with the aid of the generalized continuum hypothesis, the existence of countably incomplete, ${\beta ^ + }$-good ultrafilters on the field of all subsets of a set of (infinite) cardinality $ \beta$. Subsequently, Kunen has proved the existence of such ultrafilters, without any special set theoretic assumptions, by making use of the existence of certain families of large oscillation. In the present paper we succeed in carrying over the original arguments of Keisler to certain fields of sets associated with the homogeneous-universal (and more generally with the special) Boolean algebras. More specifically, we prove the existence of countably incomplete, a-good ultrafilters on certain powers of the a-homogeneous-universal Boolean algebras of cardinality a and on the a-completions of the a-homogeneous-universal Boolean algebras of cardinality a, where $a = a^{[unk]} > \omega$. We then develop a method that allows us to deal with the special Boolean algebras of cardinality $a = 2^{[unk]}$. Thus, we prove the existence of an ultrafilter p (which will be called adequate) on certain powers $\mathcal{S}_\alpha ^\delta $ of the special Boolean algebra $ {\mathcal{S}_\alpha }$ of cardinality a, and the existence of a specializing chain $\{ {\mathcal{C}_\beta }:\beta < \alpha \}$ for $ {\mathcal{S}_\alpha }$, such that $\mathcal{C}_\beta ^\delta \cap p$ is ${\beta ^ + }$-good and countably incomplete for $\beta < \alpha$. The corresponding result on the existence of adequate ultrafilters on certain completions of the special Boolean algebras is more technical. These results do not use any part of the generalized continuum hypothesis.
A generalization of univalent functions with bounded boundary rotation
Edward J.
Moulis
369-381
Abstract: This paper introduces a class of functions which generalizes both those functions $f(z)$ with bounded boundary rotation and those functions for which $zf'(z)$ is a-spirallike. A simple variational formula for this class is derived and used to determine sufficient conditions for the univalency of functions there in. Various representations for these functions are given, and these are used to derive another condition for univalence; this one is the best known so far in the subclass consisting of functions $f(z)$ for which $zf'(z)$ is a-spirallike. Bounds on the modulus of the Schwarzian derivative are also derived; these are sharp in the subclass of functions having bounded boundary rotation.
$Q$-composable properties, semigroups and ${\rm CM}$-homomorphisms
A. R.
Bednarek;
K. D.
Magill
383-398
Abstract: A certain type of topological property is investigated. To each such property and each topological space satisfying various conditions there is associated, in a natural way, a semigroup of relations. The nonconstant, union and symmetry preserving homomorphisms from one such semigroup into another are completely determined and this results in a topological version of the Clifford-Miller Theorem on endomorphisms of the full binary relation semigroup on a set.
Equicontinuity and indivisibility in transformation groups
Ping-fun
Lam
399-424
Abstract: A notion called indivisibility for transformation groups is studied in some detail. The proofs of some theorems announced in [10] are given. Certain types of nonequicontinuous sets which do not disconnect their metric continuum phase spaces and which were not known to exist previously are exhibited.
Rings which are almost polynomial rings
Paul
Eakin;
James
Silver
425-449
Abstract: If A is a commutative ring with identity and B is a unitary A-algebra, B is locally polynomial over A provided that for every prime p of A, $ {B_p} = B{ \otimes _A}{A_p}$ is a polynomial ring over ${A_p}$. For example, the ring $Z[\{ X/{p_i}\} _{i = 1}^\infty ]$, where $ \{ {p_i}\} _{i = 1}^\infty$ is the set of all primes of Z, is locally polynomial over Z, but is not a polynomial ring over Z. If B is locally polynomial over A, the following results are obtained, B is faithfully flat over A. If A is an integral domain, so is B. If $ \mathfrak{a}$ is any ideal of A, then $ B/\mathfrak{a}B$ is locally polynomial over $ A/\mathfrak{a}$. If p is any prime of A, then pB is a prime of B. If B is a Krull ring, so is A and the class group of B is isomorphic to the class group of A . If A is a Krull ring and B is contained in an affine domain over A, then B is a Krull ring. If A is a noetherian normal domain and B is contained in an affine ring over A, then B is a normal affine ring over A. If M is a module over a ring A, the content of an element x of M over A is defined to be the smallest ideal ${A_x}$ of A such that x is in $ {A_x}M$. A module is said to be a content module over A if ${A_x}$ exists for every x in M. M is a content module over A if and only if arbitrary intersections of ideals of A extend to M. Projective modules are content modules. If B is locally polynomial over a Dedekind domain A, then B is a content module over A if and only if B is Krull.
Valuations, primes and irreducibility in polynomial rings and rational function fields
Ron
Brown
451-488
Abstract: The set of extensions of the valuation v on a linearly compact (i.e. maximal) field F to the polynomial ring $ F[x]$ is shown to depend only on the value group and residue class field of v. By a method related to Mac Lane's construction of (rank one) valuations on polynomial rings, a determining invariant is associated with each such extension, called its ``signature". Very roughly, a signature is a pair of sequences, one in the algebraic closure of the residue class field of v and one in the divisible closure of the value group of v. Signatures are also associated with various mathematical objects by means of the extensions of the above sort which naturally arise from them. For example, the set of nonconstant monic irreducible polynomials in $F[x]$, the set of all finite Harrison primes of the polynomial ring of a global field, and the set of equivalence classes of valuations on the field of rational functions over a global field are each shown to be bijective with a simple set of signatures. Moreover, these objects are studied by means of their associated signatures. For example, necessary and sufficient conditions for irreducibility in $ F[x]$ are given, independent of the language of signatures.
Generalized eigenfunctions and real axis limits of the resolvent
N. A.
Derzko
489-506
Abstract: Let $(\mathcal{H},( \cdot , \cdot ))$ be a Hilbert space and A, E be a selfadjoint operator and corresponding spectral measure in $ \mathcal{H}\;(A = \smallint \lambda E(d\lambda ))$. It is known that for a suitable positive subspace ${\mathcal{H}_ + } \subset \mathcal{H}$ and measure $ \rho$ the generalized eigenfunctions $\displaystyle {\phi _{\lambda ,f}} = \mathop {\lim }\limits_{h \to 0} \frac{{E(... ...hop {\lim }\limits_{\Delta \to \lambda } \frac{{E(\Delta )f}}{{\rho (\Delta )}}$ exist in $ {\mathcal{H}_ - }$, the corresponding negative space, for $\rho$-almost every $\lambda$ and $f \in {\mathcal{H}_ + }$. It is shown that for each $ \lambda$ the ${\phi _{\lambda ,f}}$ form a pre-Hilbert space $ {\mathcal{H}_\lambda }$ using the natural inner product $ {({\phi _f},{\phi _g})_\lambda } = {\lim _{\Delta \to \lambda }}((E(\Delta )f,g)/\rho (\Delta ))$, and that $ \left\Vert \phi \right\Vert - \leq C{\left\Vert \phi \right\Vert _\lambda }$. Furthermore, if $\{ \phi (\lambda ,\alpha )\}$ is a suitably chosen basis for $ {\mathcal{H}_\lambda }, - \infty < \lambda < \infty$, then one obtains the eigenfunction expansion suggested by $\displaystyle (f,g) = \int {\rho (d\lambda )\;\sum\limits_{\alpha ,\beta } {(f,... ...)){\sigma _{\alpha \beta }}(\lambda )\overline{(g,\phi (\lambda ,\beta )).}} }$ . Finally it is shown that, for a suitable function $w(\varepsilon ,\lambda ),{\phi _{\lambda ,f}}$ is given by ${\lim _{\varepsilon \downarrow 0}}w(\varepsilon ,\lambda )[R(\lambda - i\varepsilon ) - R(\lambda + i\varepsilon )]f$, where $R(z) = {(z - A)^{ - 1}}$.
Erratum to: ``On character sums and power residues''
Karl K.
Norton
507