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A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

An isomorphism between certain subspaces of the Hilbert spaces of symmetric and antisymmetric n-point functions (or, more generally, symmetric and antisymmetric tensor products of a Hilbert space) is described. It permits a construction of generating functionals for sets of antisymmetric functions. In this way the theory of Hilbert spaces of functional power series as described in [7] and [8] can be extended to the case of antisymmetric coefficients. As an application, the functional representation for the anticommuntation relations is derived. It enables to obtain a functional formulation of quantum field theory also in the antisymmetric case without the use of Grassman algebras.

It is well known that for X a CW-complex, K(X) and Hev(X) are isomorphic modulo finite groups, although the "isomorphism" is not natural. The purpose of this paper is to improve this result for X a finite CW-complex. 1. Preliminaries. For the basic definitions and theory (1) of X-rings, we refer to [21, [121. The ring Z of integers has a X-ring structure with Am: Z -Z the function Xmn) (fn)n(n1) .(n-m+ 1) m ml Definition. An augmented X-ring is a X-ring R together with X-ring homomorphisms i: Z -R and e: R -*Z such that ci = 1. Since the Definition implies that i: Z -4 R is a monomorphism, we think of Z C R as the multiples of the identity. Let $3 denote the category of augmented X-rings and X-ring homomorphisms which commute with the augmentation. If B e Ob 3, we write B7y for the nth term of the y-filtration on B, and P(B) for the associated graded ring. We note that ['(B)o = Z. Let d be the category of commutative graded rings A with Ao = Z. We define A: d $ as follows: If A E Ob , then as a set A(A)= l An. If a EA(A), we denote the component in An by an and for convenience of notation define ao = 1 E AO and write a as either of the formal expressions 1 +al +a 2+ *+an + * or ?a.. oro ^10~~~~~~~~~~ If a, b E A(A), we define their sum, a @ b, componentwise: (a $ b) = i a.b. n i+j=n Z so that '@' is analogous with multiplication of formal power series. This operation makes A(A) into an Abelian group, and we denote the inverse of a by e a. In particular if a is an element with an = 0 for n > 2, then Received by the editors December 13, 1972 and, in revised form, July 9, 1973. AMS (MOS) subject classications (1970). Primary 55B15; Secondary 55B40, 55F40. (I) We are indebted to Atiyah and Tall [21 for explicit proofs, in particular those based on 'formal algebra'. Copyright

It has been shown by T. Head that if G is an abelian and S = Je~A G is a union of abelian groups, then group there exists an isomorphism e:G@S ~UacA (G@G) defined by e(g@s) = g@s for all g (G, s ~ S. In this paper it is shown that if S = UaeA Ga and T = U 8~B H8 are both unions of abelian groups then the surjective homomorphism 4: S~T ÷ Ue~AUS~B (Ga@H~) defined by ~(s ~t) = s ~t is an isomorphism if and only if either S or T is a group. This theorem is then used to show that if S is an abelian semigroup and T = ~e~A Ge is a union of at least two disjoint groups, then there exists a natural surjective homomorphism ~: S~T~e, A S~G which is an isomorphism iff S is archimedean. In this paper all semigroups will be abelian. Z and 0 will be the additive group of integers and singleton semigroup respectively. Homomorphisms will be referred to as maps. The following development of the union of groups U and the surjective homomorphism ~ in Theorem 1 are a generalization of results by T. Head [2]. Given S = ~A Ge and T = US~B HS, unions of groups

It has been conjectured for some time [St2], [D], [S-D], [D-S2] that for a semilocal ring R, the homomorphisms On:K2(n, R)-+K2(n + l, R), known to be surjective for all n^.2 [St2], [S-D], are in fact isomorphisms for n^.3. Various special cases have been proved, most notably the difficult theorem of Matsumoto for fields [Ma, Corollaire 5.11] and the case of discrete valuation rings [D-Sl]. Matsumoto also shows that 62 is not an isomorphism in general. In this note we announce the proof of the following theorem, details of which will be published elsewhere. Unexplained notation and terminology is that of [Mi].

A subset S of a group G is said to be a sum-free set if S ∩ (S + S) = ⊘ . Such a set is maximal if for every sum-free set T ⊆ G, we have |T| ⩽ |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ⊆ T ⊆ G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order.

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U f -algebra. For any U-algebraAsetU A =U i eI({i}×(A K i—domf i A )), where (K i) i eI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ⊆N such that there exists a bijective mapping σ:U F(N, U) →N −M satisfying σ((i, α)) ∉ α(K i ) for all (i, α) ∈U F (N, U), and, for the structureg=(g i)ieI defined by ,g i : =f i F(N, U) ∪ {(α, σ((i, α))) | (i, α ∈U F(N, U)} id M induces an isomorphism betweenF(M, U f ), and (F(N, U)g).

Let K ( Z p ) K({Z_p}) be the Eilenberg-Mac Lane spectrum with homotopy Z p {Z_p} and let A = H ∗ ( K ( Z p ) ; Z p ) A = {H^ \ast }(K({Z_p});{Z_p}) —the mod p \bmod p Steenrod algebra. Let X X be a locally finite spectrum. It is proven that \[ [ K ( Z p ) , X ] → Hom A ( H ∗ ( X ; Z p ) , A ) [K({Z_p}),X] \to {\operatorname {Hom} _A}({H^ \ast }(X;{Z_p}),A) \] is an isomorphism. It is also proven that there is a unique decomposition X = ( ⊕ K ( Z p ) ) ⊕ Y X = ( \oplus K({Z_p})) \oplus Y where H ∗ ( Y ; Z p ) {H^ \ast }(Y;{Z_p}) as an A A -module has no free summands.

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

Let X be a complex space and A be a subspace of X. Suppose there exists a proper surjective morphism/: A-+A', where A' is another complex space. We say that X can be blown down along /, if the following conditions are satisfied; there exists a complex space X' containing A' as a subspace, and a proper surjective morphism /': X—*X' such that i) f'(A)=A and f\A coincides with/, and ii) / gives an isomorphism of X—A and X'—A'. In this case we say that (

An analog is constructed of the spectral sequence for a fibration given over some skeleton of a cellular space. The construction is applied to construct an isomorphism of the first k terms of the spectral sequences of two fibrations with common cellular base whose restrictions to the (k−1)-skeleton of the base coincide.

Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex. 0. Introduction. In the previous paper [1] the question of whether all B-convex spaces are reflexive was discussed. The concept of a P-convex space was introduced by C. Kottman [4] as follows: Definition. For a positive integer n, let P(n, X) be the supremum of all numbers r such that there is a set of n disjoint closed balls of radius r inside U(X)= fx: llxll < 11. X is said to be P-convex if P(n, X) < Y2 for some n. Kottman showed that all P-convex spaces are both B-convex and reflexive. Therefore the question "Is there a B-convex space that is not P-convex?" is of interest. Many properties of B-convex spaces are not known for P-convex spaces. In this paper we consider two of these properties and prove partial analogs of them for P-convex spaces: The first property is that direct sums of B-convex spaces are B-convex [2]. The proof of this fact for B-convex spaces rests on the invariance of B-convexity under isomorphism, but it is not known whether P-convexity possesses this invariance. Two partial analogs of the direct sum property are obtained, Theorems 1.3 and 1.5, using Ramsey's theorem of combinatorics. The second property is that a space is B-convex if each subspace having a basis is B-convex [1]. A partial analog of this is proved, Theorem 2.1, using one of the direct sum results. We will use the following characterization of P-convexity from Remark 1.4 of [4]: Let a set of n elements be called 5 separated of order n provided the distance between any two elements of the set is at least &. Then a space X is P-convex if and only if for some positive integer n and some positive number e < 2 there is no 2 e separated set of order n in U(X). Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15, 05A05.

Gluskin [2] has shown that if α is an isomorphism of a weakly reductive semigroup S onto a semigroup T, if V is a dense extension of S and T is densely embedded in W then α extends uniquely to an isomorphism of V into W. Here we consider the problem of extending epimorphisms and as a consequence of a few simple observations obtain as the main theorem a homomorphism of Ω(S), for any semisimple semigroup S, into the product of the translational hulls of the principal factors of S. A few consequences are considered.

A complete answer is given to the question, does isomorphism of the endomorphism groups of primary Abelian groups imply isomorphism of the groups themselves ? It is presumed that the generalized continuum hypothesis is valid.

The structural pattern inherent in the genealogy of the Old Testament is delineated (1). Correlating this structural pattern with the worldview of Ancient Israel, it is hypothesized that the genealogy functions as a model for the resolution of certain conflicts within the worldview (II). Finally, an analysis of the Jacob narration demonstrates the structural isomorphism of genealogy and individual case (III).

1. Notations ~ and Lemmata Let T be a compact (Hausdorft) space, # a positive Radon measure on T with support Suppp= T, L~'~(T, p) the seminormed algebra of bounded measurable real valued functions on T, L~(T, ~) the Banach algebra LP~ (T, #) modulo #-negligible functions and f ~f the canonical mapping of c~°~(T, p) onto L~'(T, #). For a lifting r of 5P~(T, p) we define ~-~: = {rB:B E J}, where 3: = {B c= T: qS~ ~ ~(T, p)}. Furthermore let (S, v) be the hyperstonean space associated with (T,/0. Let U be the isomorphism of L~(T, IO onto L~(S, v) such that I~(f)=v(Uf) for every fccga(T). Then there exists a continuous surjection n:S~ T such that p=n(v). We denote by Q the strong lifting of L.W~(S, v) and we put 5QQ: = {QC: Cc5~}, where 50: = {C_C_ S: ~cCLP~(S, v)}. Note that 5PQ is the class of all simultaneously open and dosed subsets of S. We also consider the following diagramm of boolean homomorphisms

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group O n+1 * (K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups o n+1 * (K,f) are models of the system of axioms.

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

A topology is introduced in a logic ℒ using the set of pure states of ℒ. It is shown that ℒ, equipped with this topology, under suitable conditions, determines the division ring ℝ, ℂ or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic ℒ to be isomorphic with the projective logic of the projections in a Hilbert space over ℝ, ℂ or 2e.

The measurement of force is based on a formal law of additivity, which characterizes the effects of two or more configurations on the equilibrium of a material point. The representing vectors (resultant forces) are additive over configurations. The existence of a tight interrelation between the force vector and the geometric space, in which motion is described, depends on observations of partial (directional) equilibria; an axiomatization of this interrelation yields a proof of part two of Newton's second law of motion.The present results (which were derived from a curious and deep isomorphism between force measurement and trichromatic color measurement) yield a kind of subunit, which needs to be incorporated into more complete axiomatizations of mechanics that would fulfill the Mach–Kirchhoff program.

In this paper, we study relations in general categories. Our approach requires that these must have finite products and factorization systems. Klein [5] has obtained a condition for composition of relations to be strictly associative. Here we consider the possibility that associativity only holds up to a coherent isomorphism, in other words that the relations are the morphisms of a bicategory in the sense of Benabou [2].