One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.

A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

Each variety of lattice ordered groups determines a variety of groups, namely the variety of groups generated by the groups i n . In this paper a completely new and different correspondence between varieties of groups and varieties of lattice ordered groups is developed. It is known that the variety of representable lattice ordered groups is defined by the law z+ Λ u-1z-u = 1. Here we consider the varieties defined by laws of this form where u is restricted to lie in some fully invariant subgroup of the free group Fx on a countable set X. All the varieties considered contain the variety of representable l-groups and therefore the free group with appropriate ordering.

1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the function

is a norm on J, denoted by || • ||p.

Soit K un compact du plan complexe C, d'intérieur non vide et connexe. A(K) désigne l'algèbre uniforme des fonctions continues sur et analytiques dans . Pour chaque , on note λz la mesure harmonique sur la frontière de du point z, et on pose λ = λz0, z0 étant un point fixé de . On dit que A(K) est une algèbre de Dirichlet faible* sur (— désigne le conjugué) est faiblement* dense dans L∞(λ) (la topologie faible* sur L∞(λ) est la topologie faible pour la dualité entre L1(λ) et L∞(X)); vu les inégalités de Harnack pour les fonctions harmoniques cette propriété est indépendante de la mesure harmonique du point z0 fixé dans .

In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.

In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

In [20], T. A. Springer proved that if A is a complete discrete valuation ring with field of fractions F, residue class field of characteristic not 2, and uniformizing parameter π then any anisotropic quadratic form q over F has a unique decomposition as q = q1 ⊥ 〈π〉q2, where q1 and q2 represent only units of A, modulo squares in F (compare [14, Satz 12.2.2], [19, §4], [18, Theorem 8.9]). Consequently the binary quadratic form x2 + πy2 represents only elements in Ḟ2 ∪ πḞ2, where Ḟ2 denotes the set of nonzero squares in F. Szymiczek [21] has called a nonzero element a in a field F rigid if the binary quadratic form x2 + ay2 represents only elements in Ḟ2 ∪ aḞ2.

Let be the space of Fourier transforms of distributions with compact support, or equivalently, the space of entire functions h satisfying the growth condition

(1)

where p(z) = |Imz| + log (1 + |z|2) and A, B are constants depending only on h. A sequence with |zk| ↑ ∞ is said to be a universal interpolation sequence for if for all such that

(2)

for constants A, B independent of k, there exists such that f(zk) = ak. In this note we will consider necessary conditions for universal interpolation in and more general subspaces of the entire functions.

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocks

B(x) = {b\{x}|x ∈ b and b ∈ B}.

The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.

Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.

Let K be a finite, imaginary and abelian extension of the rational number field Q, and let M be the maximal real subfield of K. It is well known that each element of order 2 in S(K), the Schur group of K, is induced from an element of order 2 in B(M), the Brauer group of M; i.e., if D is a quaternion division algebra central over K such that its class [D] in B(K) is in fact in S(K) then [D] = [B ⊗MK] where B is a quaternion division algebra with [B] ∈ B(M). A natural question to ask is: “When is every element of S(K) of order 2 induced from S(M)?” The main result of this paper is to provide necessary and sufficient conditions for this to occur when G(L/K), the Galois group of L over K, is cyclic where L is the smallest root of unity field containing K.

This paper is a contribution to applied stability theory. Our purpose is to investigate the complexity of lattices by determining the stability of their first order theories.

Stability measures the complexity of a theory T by counting the number of different “kinds” of elements in models of T. The notion of ω-stability was introduced by Morley [26] in 1965 and generalized by Shelah [31] in 1969. Shelah classified all first order theories according to their stability properties.

Stability and -categoricity are closely related (see [26] and [1]). In fact, the notions of stable, superstable and ω-stable can be regarded as successive approximations of -categorical. -categoricity is a very strong property while stability, superstability and ω-stability facilitate the classification of more “complex” theories.

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, where

This class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).

A Steiner system S(t, k, v) is a collection of k-subsets, called blocks, of a v-set of points with the property that any t-subset of points is contained in a unique block. We assume 1 < t < k < v. A Steiner system is called block-schematic if the blocks form an association scheme with the relations determined by size of intersection. Ito and Patton [3] proved that if S(4, 5, v) is block-schematic, then v = 11. The purpose of this paper is to extend this result, and we prove the following theorem.

THEOREM. A Steiner system S(t, t + 1, v) is block-schematic if and only if one of the following holds: (i) t = 2, (ii) t = 3, v = 8, (iii) t = 4, v = 11, (iv) t = 5, v = 12.

Throughout this paper “space” means “T1 topological space.“

1. The concept of an ortho-base was introduced by W. F. Lindgren and P. J. Nyikos.

Definition 1. A base of a space X is called an ortho-base provided that for each subcollection either is open or is a local base of a point x ∊ X [17].

Ortho-bases are related to interior-preserving collections which have been known for some time.

Definition 2. A collection of open sets of a space X is called interior-preserving provided that the intersection of any subcollection is open. A space X is called orthocompact provided that each open cover has an open interior-preserving refinement.

It was proved in [17], in particular, that each space with an ortho-base is orthocompact, and each orthocompact developable space (which is the same as a non-archimedean quasi-metrizable developable space [4]) has an ortho-base.

A chaque c.w. complexe X, 1-connexe, de type fini, est associé une algèbre différentielle graduée commutative (ΛZ, d) qui est un modèle dans le sens de Sullivan [10] et représente donc le type d'homotopie rationnelle de X.

ΛZ est munie d'une seconde graduation Z = ⊕q≧0Zq. La différentielle d se décompose alors sous la forme et d0(Zp) ⊂ (Λ2Z)p–1 et di(Zp) ⊂ Zp–i, i ≧ 1. En particulier d0 est purement quadratique tandis que di est du premier degré, i ≧ 1.

L'algèbre ΛZ et la partie quadratique d0 ne dépendent que des nombres de Betti de X. Les différents types d'homotopie rationnelle de nombres de Betti donnés sont donc paramétrés par les déformations di. La partie d1 exprime la structure multiplicative de l'algèbre de cohomologie rationnelle de X, les parties d2, d3, d4. … les structures algébriques plus complexes du type d'homotopie rationnelle.

R. Matsuda has shown that a group ring is a Krull domain if and only if the coefficient ring is a Krull domain and the group is a torsion-free abelian group satisfying the ascending chain condition (ace) on cyclic subgroups [6]. D. F. Anderson has used this to obtain a partial determination of when a semigroup ring is a Krull domain, and under certain circumstances to describe the divisor class group of such a ring ([1], [2]). Using some of Anderson's techniques, but taking a different approach, we arrive at a complete answer of a different nature to these questions. We call a semigroup satisfying the major new conditions arising a Krull semigroup, and define its divisor class group.

In particular, every abelian group is the divisor class group of such a ring, and it follows that every abelian group is the divisor class group of a quasi-local ring, which seems to be a new result.

Let G be an amenable locally compact group. We will use the terminology of [3] and denote by VN(G) the Von Neumann algebra of the regular representation and by A(G) its predual, which is the algebra of the coefficients of the regular representation. The Von Neumann algebra VN(G) is, in a natural fashion, a module with respect to A(G) [3].

The algebra of bounded linear operators on VN(G), which commute with the action of A(G), has been studied in [6] and in [1]. If UCB(Ĝ) is the space of the elements of VN(G) of the form vT, for some v in A(G) and some T in VN(G) (see for instance [4]), in [6] and in [1] it is proved that, for any amenable locally compact group there exists an isometric bijection between and UCB(Ĝ)*.

In [8] Jha raised the following problem.

(*) Let Γ be a spread whose components are subspaces of V2n(GF(q)). Suppose G ≦ Aut Γ leaves a set of q + 1 components invariant while acting transitively on Γ\Δ.

Find the possibilities for Γ or, more generally, the possibilities for (G, Γ, n, q).

Many special cases of (*) have been settled. For instance, Cohen et al [1] have shown that if G fixes two non-zero points of V, that do not both lie in the same component of Γ, then Γ is the spread associated with either a Hall plane or the Lorimer-Rahilly plane of order 16 (LR-16) [14], [18].

Another such result is given in [8]; there it is shown that if q is a prime number and G is a one-dimensional projective unimodular group then Γ is the spread associated with one of the following translation planes:

• (1) the Desarguesian planes of order 4, 8, or 9;

• (2) the nearfield plane of order 9;

• (3) LR-16;

• (4) the translation plane JW-16, obtained by transposing the slope maps of LR-16 [19].

Consider an arbitrary set X and an arbitrary disjunctive lattice of subsets of X, ℒ. The algebra of subsets of X generated by ℒ is denoted by , the set of all ℒ-regular measures on , by MR(ℒ), and the associated Wallman space, a compact T1 space, by IR(ℒ); assume X is embedded in IR(ℒ) (otherwise, consider the image of X in IR(ℒ)).

In part of an earlier paper [4] the work of Knowles [15] and Gould and Mahowald [11] was generalized from the explicit topological setting of X, a Tychonoff space, with ℒ the lattice of zero sets of X, to the above setting, with the added assumption that ℒ was also δ and normal. This was done so that the important Alexandroff Representation Theorem [1] could be utilized in order to induce two associated measures , and defined on and respectively, where W(ℒ) is the Wallman lattice in IR(ℒ).