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We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) E n ! \mathcal {E}_n^! of the Fomin–Kirillov algebras E n \mathcal {E}_n ; these algebras are connected N \mathbb {N} -graded and are defined for n ≥ 2 n \geq 2 . We establish that the algebra E n ! \mathcal {E}_n^! is module finite over its center (and thus satisfies a polynomial identity), is Noetherian, and has Gelfand–Kirillov dimension ⌊ n / 2 ⌋ \lfloor n/2 \rfloor for each n ≥ 2 n \geq 2 . We also observe that E n ! \mathcal {E}_n^! is not prime for n ≥ 3 n \geq 3 . By a result of Roos, E n \mathcal {E}_n is not Koszul for n ≥ 3 n \geq 3 , so neither is E n ! \mathcal {E}_n^! for n ≥ 3 n \geq 3 . Nevertheless, we prove that E n ! \mathcal {E}_n^! is Artin–Schelter (AS-)regular if and only if n = 2 n=2 , and that E n ! \mathcal {E}_n^! is both AS-Gorenstein and AS-Cohen–Macaulay if and only if n = 2 , 3 n=2,3 . We also show that the depth of E n ! \mathcal {E}_n^! is ≤ 1 \leq 1 for each n ≥ 2 n \geq 2 , conjecture that we have equality, and show that this claim holds for n = 2 , 3 n =2,3 . Several other directions for further examination of E n ! \mathcal {E}_n^! are suggested at the end of this article.

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

We construct curve counting invariants for a Calabi-Yau threefold Y Y equipped with a dominant birational morphism π : Y → X \pi :Y \to X . Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when π : Y → Y \pi :Y\to Y is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in the case when Y Y is a crepant resolution of X X , the coarse space of a Calabi-Yau orbifold X \mathcal {X} satisfying the hard Lefschetz condition. In this case, our partition function is equal to the Pandharipande-Thomas partition function of the orbifold X \mathcal {X} . Our methods include defining a new notion of stability for sheaves which depends on the morphism π \pi . Our notion generalizes slope stability which is recovered in the case where π \pi is the identity on Y Y .

Three questions about the intertwining operators for the generalized principal series on a symmetric R R -space are solved: description of the functional kernel, both in the noncompact and in the compact picture, domain of convergence, and meromorphic continuation. A large use is made of the theory of positive Jordan triple systems. The meromorphic continuation of the intertwining integral is achieved via a Bernstein-Sato identity, and a precise description of the poles is obtained.

In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ( { 1 } a , c , { 1 } b ) , (\{1\}^a,c,\{1\}^b), ( { 2 } a , c , { 2 } b ) (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p . p. The congruences obtained allow us to find nice p p -analogues of Leshchiner’s series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 7 and 9 9 modulo p p . As a further application we provide a new proof of Zagier’s formula for ζ ∗ ( { 2 } a , 3 , { 2 } b ) \zeta ^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.

We prove that two countable locally finite-by-abelian groups G , H G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely generated or both are infinitely generated. On the other hand, we show that each countable group G G that coarsely embeds into a countable abelian group is locally nilpotent-by-finite. Moreover, the group G G is locally abelian-by-finite if and only if G G is undistorted in the sense that G G can be written as the union G = ⋃ n ∈ ω G n G=\bigcup _{n\in \omega }G_n of countably many finitely generated subgroups such that each G n G_n is undistorted in G n + 1 G_{n+1} (which means that the identity inclusion G n → G n + 1 G_n\to G_{n+1} is a quasi-isometric embedding with respect to word metrics on G n G_n and G n + 1 G_{n+1} ).

Let f be a germ of a holomorphic self-map of C 2 at the origin O tangent to the identity, and with O as a nondicritical isolated fixed point. A parabolic curve for f is a holomorphic f-invariant curve, with O on the boundary, attracted by O under the action of f. It has been shown by M. Abate (2001) that if the characteristic direction [v] ∈ P(T O C 2 ) has residual index not belonging to Q + , then there exist parabolic curves for f tangent to [v]. In this paper we prove, using a different method, that the conclusion still holds just assuming that the residual index is not vanishing (at least when f is regular along [v]).

Let A be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence c n (A) is asymptotic to a function of the form an g l n , where l ∈ N and g ∈ Z.

Two permutations of [n] are comparable in the Bruhat order if one is closer, in a natural way, to the identity permutation, 12 ... n, than the other. We show that the number of comparable pairs is of order (n!) 2 /n 2 at most, and (n!) 2 (0.708)" at least. For the related weak order, the corresponding bounds are (n!) 2 (0.362)" and (n!) 2 Π n i=1 (H (i) /i), where H (i):= Σ i j=1 1/j. In light of numerical experiments, we conjecture that for each order the upper bound is qualitatively close to the actual number of comparable pairs.

Let S S be a complex non-singular projective surface of general type with a genus 2 2 fibration and χ ( O S ) ≥ 5 \chi (\mathcal O_S)\geq 5 . Let G ⊂ Aut ⁡ S G \subset \operatorname {Aut}S be a non-trivial subgroup of automorphisms of S S , inducing trivial actions on H i ( S , Q ) H^i(S,\mathbb {Q}) for all i i . Then | G | = 2 |G|=2 , K S 2 = 4 χ ( O S ) K_S^2=4\chi (\mathcal O_S) and q ( S ) = 1 q(S)=1 . Examples of such surfaces are given.

In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for ∏ n = 1 ∞ ( 1 − q n ) 10 \prod _{n=1}^\infty (1-q^n)^{10} is given. The proofs are self-contained and elementary.

Let ϕ \phi be a hyperbolic diffeomorphism on a basic set Λ \Lambda and let G G be a connected Lie group. Let f : Λ → G f : \Lambda \rightarrow G be Hölder. Assuming that f f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G u : \Lambda \rightarrow G is a measurable solution to f = u ϕ ⋅ u − 1 f=u\phi \cdot u^{-1} a.e., then u u must in fact be Hölder. Under an additional centre bunching condition on f f , we show that if f f assigns ‘weight’ equal to the identity to each periodic orbit of ϕ \phi , then f = u ϕ ⋅ u − 1 f = u\phi \cdot u^{-1} for some Hölder u u . These results extend well-known theorems due to Livšic when G G is compact or abelian.

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If K C M are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight T: M?+ N + if and only if there exist fns weights b on M and cp on K satisfying r't (x) = oa (x) Vx E K, t E R. In fact, T can be chosen such that f = p o T; T is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory. 0. INTRODUCTION Consider the following theorem of Haagerup: Theorem. Let KV c M be von Neumann algebras. There exists a faithful, semifinite normal operator valued weight T: M+ -S K+ if and only if there exist faithful semi-finite normal weights (p on M and (p on KV such that a,' (x) = UCf (x), x E J\r. If this condition is satisfied, then T can be chosen in such a way that 0 = 0 o T; moreover, T is uniquely determined by this identity. Haagerup's proof of the above depends upon the use of structure theory, i.e., the existence of a crossed product decomposition of a general von Neumann algebra [Haal], [Haa2]. However, it is possible to demonstrate this without recourse to the crossed product. The method for showing this depends critically on a result of Connes-Masuda which provides a converse to the theorem purporting the existence of the Radon-Nikodym cocycle derivative. (Masuda's result does not use structure theory, unlike that of Connes.) This result implies that, given an invariance condition like that in the hypothesis, a weight on KV induces one on M. With the ability to associate weights on JKto weights on M, it is possible to construct an operator valued weight T: M+ -+ K+. The desire to recast the proof of the above in a form which does not depend on any structure theory is motivated by the belief that it is a result which is fundamental in the theory of non-commutative integration. Specifically, application of an operator valued weight to a positive element of M (which results in an element in Received by the editors January 30, 1997. 1991 Mathematics Subject Classification. Primary 46L50; Secondary 22D25. This work is supported, in part, by NSF Grant DMS95-00882. ?1999 American Mathematical Society 323 This content downloaded from 157.55.39.27 on Wed, 07 Sep 2016 05:33:38 UTC All use subject to http://about.jstor.org/terms 324 TONY FALCONE AND MASAMICHI TAKESAKI the extended positive cone of A) can and should be thought of as "partial integration." The "space" over which we are integrating is not always explicit; however, just as we often think of von Neumann algebras as "non-commutative L?-spaces," we sense the presence of an underlying measure space indirectly, by observing the interactions of function-like objects defined on this "space." Moreover, this point of view allows us to interpret the expression = o T as a form of Fubini's theorem (or more precisely, Tonelli's theorem). Because we believe that structure theory should, in some sense, be a consequence of the results of integration theory, and not vice-versa, a proof which does not involve the crossed product is indicated. It should also be noted that work in this area was done by Hirakawa [Hir]. The work presented herein addresses the problem from a different direction than he did, however, and therefore represents a new approach. 1. SOME PRELIMINARIES In order to discuss the subject of operator valued weights, we begin with a review of some of the relevant terms and concepts. When studying weights, we consider the extended positive real numbers R+ U {oo}. To study "unbounded conditional expectations" (i.e., operator valued weights), we need to consider the "extended positive part" MAf+ of the von Neumann subalgebra H of M. We begin with the following: Definitlion 1.1. For a von Neumann algebra M, the extended positive cone M+ of M is the set of maps m: +[0, oo] with the following properties: (i) m(A(p) = Am((p), p E MA+ A > 0, (iii) m is lower semi-continuous. Clearly, the positive part M+ of M is a subset of M+. It is easy to see that M+ is closed under addition, multiplication by non-negative scalars and increasing limits. Example 1.2. Let {M, S} be a von Neumann algebra and A a positive selfadjoint operator on Si affiliated with M. Suppose that A j / A de(A) is the spectral decomposition of A. For each S? e M+, put mA (.p) A d(p(e(A)) Then mA satisfies the conditions (i), (ii) and (iii) of Definition 1.1. The last condition, the lower semi-continuity, follows from n mA((o) = sup ,o(A,) with AnjAde(A) e M+. n It now follows that mA(wa) A d(e-(A), = A /241121 E?(A,1/2) If B is another positive self-adjoint operator on Sj affiliated with M, then the equality mA mB means precisely A = B. Hence the map A F mA C M+ is This content downloaded from 157.55.39.27 on Wed, 07 Sep 2016 05:33:38 UTC All use subject to http://about.jstor.org/terms OPERATOR VALUED WEIGHTS WITHOUT STRUCTURE THEORY 325 injective. Thus, the set of positive self-adjoint operators affiliated with M can be identified with a subset of the extended positive cone M+. Definition 1.3. For m, n E M+, A > 0 and a E M, we define the following operations: (Am)p) :-Am((p), EC M*+, (Tm + n) ((p) Tn m(() + n ((p), fo E M*+, (a*ma)((p) :-m(a(pa*), ,o E M*. We also note here that supi mi of an increasing net in M+ can be naturally defined. Lemma 1.4. Let {M,5i} be a von Neumann algebra. To each m E M+, there corresponds uniquely a pair {A, A} of a closed subspace A of S) and a positive selfadjoint operator A on A such that (i) A is affiliated with M, in the sense that the projection to A belongs to M, and A is affiliated with M; (ii) (1) ~ ~ ~~~ m(Wt,) IIIA1/(l E: E-(A/2 1 +oo, otherwise. Here wz means, of course, the functional x C M A4 (x (x ). The proof of the above lemma, as well as the next theorem and its corollary, is standard, and is therefore omitted. (See, for instance, [Haal].) We say that an element m E M+ is semi-finite if { E M+ : m((p) 0 for every non-zero o E M*. Theorem 1.5. Let M be a von Neumann algebra. Each m E M+ has a unique spectral decomposition of the form (2) m(A) j A\d(p(e(A)) + oo(p(p), EM+, where {e(A): A E R+} is an increasing family of projections in A/I which is ostrongly continuous from the right, and p = -limAOe(A). Furthermore, e(O) = 0 if and only if m is faithful, and p 0 O if and only if m is semi-finite. To simplify the notation, we write (2') m-H+oop, H= A de (A), when m has the form of (2). We keep the convention 0 (+oo) = 0. Although we consider H as an operator affiliated with M, we use the following notation: D(H1/2) = {d E : m(WS) O, m cM+, o(m?+rn) =z((m)-+Vp(n), m,n E M+, W(sup mi) = sup W(mi) i i for any increasing net {mi} in M+, where we say the net {mi} is increasing if {mi(w)} is increasing for every w c M+. So, we may think of the extended positive cone of M as M+, along with a myriad of "points at infinity" adjoined to it. There will be a different infinite point for each projection in M. When we think about the extended positive cone in this way, it is clear why we adopt the notation in (2'). Now, for each m = H + oop, we put mno =(I + H)-,(l-P), m? = H(1 + H)I(1-p) + -p > Or We note that both mo and m, are bounded. Lemma 1.7. (i) For each m, n A M+, we have the following equivalence: m no X~ m? 0. (ii) Let {mi} H= Hi + oopi} be an increasing net in M+ and m = H + oop. Then we have mi / m X (mi)o \ mO X (mi)/ m/ , ? > 0. Proof. (i) Suppose m -14) so that ((Ei + K)-1) |) ((I + K)-141 ) S5 Setting (1 + K)-'1 = 0 for ( c ?' n i, we view (1 + K)-1 as an operator on A and have (1 + H)-1 > (1 + K)-1. Then we have (1 + H)-1/2 > (1 + K)-1/2, and D(H1/2) (1 + H)--1/2.q D (1 + K)'/2-q =D(K1/2) The argument in the first paragraph shows that (1 + H)(1 + E(1 + H)- lim 11 (1 + H)/2(1 + E(1 + H)-1)-1/2(112 =11 (1 + H) 1/2( 112 so we conclude that 1 + n > 1 + rn; equivalently n > m. Now we have seen the equivalence: m no. For a fixed E > 0, we have then m (En)o X 1 (Em)o 1(En)o X m = -(1 (em)0) < --(1E-(n)o) = n(ii) By (i), the net {(mj)o} is decreasing. If = infi(mi)o, then there exists n c AM+ such that no = X, because (mi)o < 1 implies e < 1. If m = supi mi, then we have mo < (mj)o, so mo ? no, which implies n < m by (i). Hence mo = infi(mi)o = lim(mi)o. Thus we prove the equivalence: mi / m X (mi)o \i mO. Finally, the equality

We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. (1.1 f -Qu = Aue' , u>O in Q, u-0 on 9Q where Q is a unit disk in R12 and A denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies IVU12dx 47r, A\ 0. Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as u (ymX )-u m)-+ 2 log asAm\O locally uniformly in x E R1R2 We also show some extensions of the above results for general two dimensional domains.

A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.

An associative ring R R is said to have stable range 1 1 if for any a a , b ∈ R b \in R satisfying a R + b R = R aR + bR = R , there exists y ∈ R y \in R such that a + b y a + by by is a unit. The purpose of this note is to prove the following facts. Theorem 3 3 : An exchange ring R R has stable range 1 1 if and only if every regular element of R R is unit-regular. Theorem 5 5 : If R R is a strongly π \pi -regular ring with the property that all powers of every regular element are regular, then R R has stable range 1 1 . The latter generalizes a recent result of Goodearl and Menal [ 5 5 ].

We investigate the properties of Lie algebras of characteristic p p which satisfy the Engel identity x y n = 0 x{y^n} = 0 for some n > p n > p . We establish a criterion which (when satisfied) implies that if a a and b b are elements of an Engel- n n Lie algebra L L then a b n − 2 a{b^{n - 2}} generates a nilpotent ideal of L L . We show that this criterion is satisfied for n = 6 , p = 7 n = 6,\,p = 7 , and we deduce that if G G is a finite m m -generator group of exponent 7 7 then G G is nilpotent of class at most 51 m 8 51{m^8} .

There is a beautiful cubic analogue of Jacobi’s fundamental theta function identity: θ 3 4 = θ 4 4 + θ 2 4 \theta _3^4 = \theta _4^4 + \theta _2^4 . It is \[ ( ∑ n , m = − ∞ ∞ q n 2 + n m + m 2 ) 3 = ( ∑ n , m = − ∞ ∞ ω n − m q n 2 + n m + m 2 ) 3 + ( ∑ n , m = − ∞ ∞ q ( n + 1 3 ) 2 + ( n + 1 3 ) ( m + 1 3 ) + ( m + 1 3 ) 2 ) 3 . {\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m^2}}}} } \right )^3} = {\left ({\sum \limits _{n,m = - \infty }^\infty {{\omega ^{n - m}}{q^{{n^2} + nm + {m^2}}}} } \right )^3} + {\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{{(n + \frac {1}{3})}^2} + (n + \frac {1}{3})(m + \frac {1}{3}) + {{(m + \frac {1}{3})}^2}}}} } \right )^3}. \] Here ω = exp ⁡ ( 2 π i / 3 ) \omega = \exp (2\pi i/3) . In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.

A self-similar measure on R n {{\mathbf {R}}^n} was defined by Hutchinson to be a probability measure satisfying ( ∗ ) ({\ast }) \[ μ = ∑ j = 1 m a j μ ∘ S j − 1 \mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}} \] , where S j x = ρ j R j x + b j {S_j}x = {\rho _j}{R_j}x + {b_j} is a contractive similarity ( 0 > ρ j > 1 , R j (0 > {\rho _j} > 1,{R_j} orthogonal) and the weights a j {a_j} satisfy 0 > a j > 1 , ∑ j = 1 m a j = 1 0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1} . By analogy, we define a self-similar distribution by the same identity ( ∗ ) ( {\ast } ) but allowing the weights a j {a_j} to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to ( ∗ ) ( {\ast } ) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F F denotes the Fourier transformation of a self-similar distribution of compact support, let \[ H ( R ) = 1 R n − β ∫ | x | ≤ R | F ( x ) | 2 d x , H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,} \] where β \beta is defined by the equation ∑ j = 1 m ρ j − β | a j | 2 = 1 \sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1} . If ρ j ν j = ρ \rho _j^{{\nu _j}} = \rho for some fixed ρ \rho and ν j {\nu _j} positive integers we say the { ρ j } \{ {\rho _j}\} are exponentially commensurable. In this case we prove (under some additional hypotheses) that H ( R ) H(R) is asymptotic (in a suitable sense) to a bounded function H ~ ( R ) \tilde H(R) that is bounded away from zero and periodic in the sense that H ~ ( ρ R ) = H ~ ( R ) \tilde H(\rho R) = \tilde H(R) for all R > 0 R > 0 . If the { ρ j } \{ {\rho _j}\} are exponentially incommensurable then lim R → ∞ H ( R ) {\lim _{R \to \infty }}H(R) exists and is nonzero.

We study varieties of rings with identity that satisfy an identity of the form x y = y p ( x , y ) xy = yp(x,y) , where every term of the polynomial p p has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let V \mathcal {V} be such a variety. The subdirectly irreducible rings in V \mathcal {V} are shown to be finite local rings and are completely described. This results in structure theorems for the rings in V \mathcal {V} and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that V \mathcal {V} also satisfies an identity of the form x y = q ( x , y ) x xy = q(x,y)x . Analogous results are shown to hold for varieties satisfying x y = q ( x , y ) x xy = q(x,y)x .

In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or de- termined up to /-isomorphism. In the group of integers Z with the usual order

We give a geometric interpretation of the quasi-projective dimension of groups, a notion introduced by Howie and Schneebeli [H-Si] as a generalization of the Identity Property.

We produce exact cubic analogues of Jacobi’s celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is \[ a n + 1 := a n + 2 b n 3 and b n + 1 := b n ( a n 2 + a n b n + b n 2 3 ) . 3 {a_{n + 1}}: = \frac {{{a_n} + 2{b_n}}} {3}\quad {\text {and}}\quad {b_{n + 1}}: = \sqrt [3]{{{b_n}\left ( {\frac {{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right ).}} \] The limit of this iteration is identified in terms of the hypergeometric function 2 F 1 ( 1 / 3 , 2 / 3 ; 1 ; ⋅ ) {}_2{F_1}(1/3,2/3;1; \cdot ) , which supports a particularly simple cubic transformation.

To each continuous semigroup of ∗ * -endomorphisms α \alpha of B ( H ) \mathfrak {B}\left ( \mathfrak {H} \right ) with an intertwining semigroup of isometries there is associated a ∗ * -representation π \pi of the domain O ( δ ) \mathfrak {O}(\delta ) of the generator of α \alpha . It is shown that the Arveson index d ∗ ( α ) {d_ * }(\alpha ) is the number of times the representation π \pi contains the identity representation of O ( δ ) \mathfrak {O}(\delta ) . This result is obtained from an analysis of the relation between two semigroups of isometries, U U and S S , satisfying the condition S ( t ) ∗ U ( t ) = e − λ t I S{(t)^*}U(t) = {e^{ - \lambda t}}I for t ≥ 0 t \geq 0 and λ > 0 \lambda > 0 .

Let H n ( F ) {H_n}(F) denote the subspace of symmetric matrices of M n ( F ) {M_n}(F) , the full matrix algebra with coefficients in a field F F . The subspace H n ( F ) ⊂ M n ( F ) {H_n}(F)\subset {M_n}(F) does not have any polynomial identity of degree less than 2 n 2n . Let \[ T k i ( x 1 , … , x k ) = ∑ α ∈ S k 1 ≤ i ≤ k , σ − 1 ( i ) ≡ 1 , 2 mod ⁡ 4 ( − 1 ) σ x σ ( 1 ) x σ ( 2 ) ⋯ x σ ( k ) , T_k^i({x_1}, \ldots ,{x_k}) = \sum \limits _{\begin {array}{*{20}{c}} {\alpha \in {\mathcal {S}_k}} \\ {1 \leq i \leq k,{\sigma ^{ - 1}}(i) \equiv 1,2\operatorname {mod} 4} \\ \end {array} } {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}}, \] , and e ( n ) = n e(n) = n if n n is even, n + 1 n + 1 if n n is odd. For all n ≥ 1 , T 2 n i n \geq 1,T_{2n}^i is an identity of H n ( F ) {H_n}(F) . If the characteristic of F F does not divide e ( n ) ! e(n)! and if n ≠ 3 n \ne 3 , then any homogeneous polynomial identity of H n ( F ) {H_n}(F) of degree 2 n 2n is a consequence of T 2 n i T_{2n}^i . The case n = 3 n = 3 is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.

We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

Using the notion of inductive proper q − 1 − LCC q - 1 - {\text {LCC}} introduced in this note, we will prove the following theorems. Theorem 1. Let M M be an R ∞ {R^\infty } -manifold and let H : X × I → M H:X \times I \to M be a homotopy such that H 0 {H_0} and H 1 {H_1} are R ∞ {R^\infty } -deficient embeddings. Then, there is a homeomorphism F F of M M such that F ∘ H 0 = H 1 F \circ {H_0} = {H_1} . Moreover, if H H is limited by an open cover α \alpha of M M and is stationary on a closed subset X 0 {X_0} of X X and W 0 {W_0} is an open neighborhood of \[ H [ ( X − X 0 ) × I ] i n M , H[(X - {X_0}) \times I] \quad {in\;M,} \] then we can choose F F to also be St 4 ⁡ ( α ) \operatorname {St}^4(\alpha ) -close to the identity and to be the identity on X ˙ 0 ∪ ( M − W 0 ) \dot X_{0} \cup (M - {W_0}) . Theorem 2. Every closed, locally R ∞ ( Q ∞ ) {R^\infty }({Q^\infty }) -deficient subset of an R ∞ ( Q ∞ ) {R^\infty }({Q^\infty }) -manifold M M is R ∞ ( Q ∞ ) {R^\infty }({Q^\infty }) -deficient in M M . Consequently, every closed, locally compact subset of M M is R ∞ ( Q ∞ ) {R^\infty }({Q^\infty }) -deficient in M M .

We apply finite operator techniques due to G. C. Rota to a combinatorial identity, which counts a collection of generalized restricted growth functions in two ways, and obtain a q-analog of Charlier polynomials and Dobinski’s equality for the number of partitions of an n-set. Our methods afford a unified proof of certain identities in the combinatorics of finite dimensional vector spaces over GF ( q ) {\text {GF}}(q) .

A K K -group G G with identity e e is said to be generated by the K K -space X X if X X is a subspace of G G containing e , X e,X algebraically generates G G , and the canonical morphism from the Graev free K K -group over ( X , e ) (X,e) on-to G G is a quotient morphism. An internal characterization of the topology of such a group G G is obtained, as well as a sufficient condition that a subgroup H H of G G be generated by a subspace Y Y of H H . Several illuminating examples are provided.

Let C ( X ) C(X) be the space of continuous functions on a compact T 2 {T_2} -space X X where each point of X X is a G δ {G_\delta } . If F : B → C ( X ) F:B \to C(X) is a biholomorphic (in the sense that F F and F − 1 {F^{ - 1}} are Fréchet differentiable) map of B = { f | ‖ f ‖ > 1 } B = \{ f\left | {\left \| f \right \|} \right . > 1\} onto a convex domain with D F ( 0 ) = I DF(0) = I , then F F is Lorch analytic (i.e., D F ( f ) ( g ) = a f g DF(f)(g) = {a_f}g for some a f ∈ C ( X ) ) {a_{f}} \in C(X)) ). Let R R be a commutative Banach algebra with identity such that the Gelfand homomorphism of R R into C ( M ) C(\mathcal {M}) is an isometry. Starlike, convex, close-to-convex, spirallike and Φ \Phi -like functions are defined in B = { x ∈ R | ‖ x ‖ > 1 } B = \{ x \in R\left | {\left \| x \right \|} \right . > 1\} for L L -analytic functions in B B and they are related to associated complex-valued holomorphic functions in Δ = { z ∈ C | | z | > 1 } \Delta = \{ z \in \left . \mathbf {C} \right |\left | z \right | > 1\} .

We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition ( a , a , b ) = λ [ a , [ a , b ] ] (a,a,b) = \lambda [a,[a,b]] . We show that any semiprime algebra over a field of characteristic ≠ 2 \ne 2 , ≠ 3 \ne 3 which satisfies the right alternative law and the above identity with λ ≠ 0 \lambda \ne 0 is a subdirect sum of (associative and commutative) integral domains.

In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x + y x+y , multiplication x ⋅ y x \cdot y , and two-place exponentiation x y x^y . For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable x x , as x → ∞ x \to \infty , are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.

Let Q Q be a set of primes that has relative density δ \delta among the primes, and let ϕ ( x , y , Q ) \phi (x,\,y,\,Q) be the number of positive integers ⩽ x \leqslant x that have no prime factor ⩽ y \leqslant y from the set Q Q . Standard sieve methods do not seem to give an asymptotic formula for ϕ ( x , y , Q ) \phi (x,\,y,\,Q) in the case that 1 2 ⩽ δ > 1 \tfrac {1}{2} \leqslant \delta > 1 . We use a method of Hildebrand to prove that \[ ϕ ( x , y , Q ) ∼ x f ( u ) ∏ p > y p ∈ Q ( 1 − 1 p ) \phi (x,y,Q) \sim x f(u) \prod _{\substack {p > y\\p \in Q}} {\left ( {1 - \frac {1}{p}} \right )} \] as x → ∞ x \to \infty , where u = log ⁡ x log ⁡ y u = \frac {{\log x}}{{\log y}} and f ( u ) f(u) is defined by \[ u δ f ( u ) = { e γ δ Γ ( 1 − δ ) , a m p ; 0 > u ⩽ 1 , e γ δ Γ ( 1 − δ ) + δ ∫ 0 u − 1 f ( t ) ( 1 + t ) δ − 1 d t , a m p ; u > 1. {u^\delta }f(u) = \left \{ {\begin {array}{*{20}{c}} {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}},} \hfill & {0 > u \leqslant 1,} \hfill \\ {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}} + \delta \int _0^{u - 1} {f(t){{(1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end {array} } \right . \] This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where Q Q consisted of all primes.

The groups in which every two conjugate operators are cominutative have recently been considered by BURNSIDE.t In the first section of the present paper, the converse limitation is imposed on a group of operations. It is assumed that every two commutative operations are conjugate, provided neither is identity, t and the groups which are possible under this hypothesis are determined. It results that the group of order 2 ? and the symmetric group of order 6 are the only grou.ps which have the property in question. In sections 2-6, somewhat similar but smaller limitations are imposed on the group. The condition is imposed in sections 2-5 that every two commutative operations of the same order are conjugate, and in section 6 that every two commutative operations (identity excluded) are so related that each of them is conjugate to some power of the other. Some of the chief properties of the groups which are possible under these limitations are derived. The sections 7-8 deal with problems closely related to the precedilng. If it is assumed that a certain simple relation exists between the number X of complete sets of conjugate operations, and the number n of distinct prime factors in the order of the group, certain commutative operations are conjugate. Much use is made of this fact in showing what groups are possible under the given hypothesis. The symbol Ga (S1= 1), S2, S3, * , * g will be used to represent the group of order g under consideration, and p1, P2, * p Pn to represent distinct primes in ascending order of magnitude so that gp a1pj2. pa-.

The point of departure for this paper is furnished by the article of Professor F. MORLEY t in the April number of the T r a n s a c t i o n s. For the convenience of the reader, the notation of that memoir has been as much as possible followed; but it will be perceived that even where, in the opening sections of the present essay, the consequent resemblance to portions of the former rises to the point of identity of formulae, the geometric meaning which underlies these is quite distinct, while in the later portions of the article it has been necessary to find forms of statement unlike those suited to the preceding work. In the last section of the article quoted (p. 184) its author points out that the problems to which that memoir is mainly devoted arise from an initial combination of n lines by pairs, while a grouping by threes, fours, or higher numbers is possible. The present paper is concerned with the case in which the lines are originally grouped in threes, and has for its basic element, analogous to the intersection of two lines as treated in the former article, the center of a circle tangent to the three. I shall briefly indicate a new series of theorems which thus arises from an altered interpretation of formulhe practically identical with those of Professor MORLEY, including analogues to his own theorems upon center-circles, as well as to the chain of propositions associated with the names of STEINER, MIQUEL, KANTOR, and CLIFFORD; I shall then develop a relation by which the new theorems are connected with those of the foregoing case; and finally I shall devote some space to the simpler aspects of the added multiplicity of forms resulting from that character by which the case here treated is chiefly distinguished from the preceding, to wit, the assignment to each line of a definite direction, the reversal of which in any instance, while leaving the original configuration of n lines apparently unaffected, entirely changes the diagram of circles built upon it.

The self-adjoint elements of a B* algebra A may be regarded as a partially ordered (real) vector space H(A), taking as positive those elements which can be written in the form x*x, for some x in A. From the standpoints both of mathematics and of physics, it is desirable to know the extent to which A is determined by H(A). Ideally, one would like to know, given a partially ordered real vector space H, exactly which B* algebras A have H(A) isomorphic to H with respect to both linear and order structure. This is a complicated question, which we will not discuss; instead we consider the simpler question: Given a B* algebra A with identity e, for what other B* algebras A1 with identity e1, is H(A) order isomorphic(2) with H(A1) under a map taking e onto e1? (The restriction involving identities is necessary if the answer is to be at all simple.) This problem has been considered by Kadison who obtained the following result [2, Theorem 10]: THEOREM. Let A and A1 be weakly closed algebras of operators with identities e and el, respectively. Let 0 be an order isomorphism(2) of H(A) onto H(A1) taking e onto el. Then the linear extension of 0 to O: A -A1

Section I of the following paper, though using the postulational method, is motivated by the consideration of classes of vectors, on a finite range PI (1, 2, ...* n), whose elements belong to a general division algebra or, as we shall say, number system. Section II deals only with vectors oni a finite range. Section I is also of use as giviing a general basis preliminiary to tlle more intensive study of (a) classes of vectors on a general range, (b) number systems over a division numbersystem; that is, to the initiation of a theory analogous to that of an "algebra over a field," where the field is replaced by an associative division number system. Notation. Throughout the paper certain logical notationst will be used as follows: logical idenitity t logical diversity definitional identity definitional identity between statements implies is equivalent to .3. such that 31 there exists is unique, used before the element which is unique: thus, I a means a is unique. and U or not . : .: :: etc. punctuation signs; the principal implication of a sentence has its signi accompanied by the largest number of periods, thus a :): b .). c is a statement that a implies that (b implies c) whereas a.). b :): c states that the implication a implies b, implies the fact c. We may also use punctuation to show continued implication, thus a.). b .). c means a.). b and b .). c.

A general notion of variation of functions on an arbitrary commutative semigroup with identity is introduced. The concept includes Hausdorff’s for the additive semigroup of nonnegative integers as well as the more recent notions introduced for semilattices. An abstract moment problem is formulated and solved.

If a subspace W of ?* is closed with respect to this new multiplication, then W is a Jordan algebra relative to the new multiplication. A special Jordan algebra(2) is a Jordan algebra isomorphic to one obtained from an associative algebra in the above manner. The title of this paper is the same as that of a paper by A. A. Albert [3]. However, in that paper Albert considers only special Jordan algebras, while we prove the principal theorem here for the "general" Jordan algebras defined by commutativity and the identity (1.1). In each case the base field is assumed to be of characteristic 0. Considering the Jordan algebra Q3, we define(3)

The principal question with which this paper is concerned is the following: if a manifold M admits a continuous associative multiplication with identity and no other idempotents, is it a group? We are able to show that in the case Al is the line or plane, the answer is in the affirmative. However, the general problem remains open. We show that in any case, there is a maximal, connected, open subgroup G of M containing the identity. This then determines a subspace L, the boundary of G in M, which we show is an ideal of G- (the closure of G). Thus, a study of M necessarily involves a thorough study of L. We conjecture, for example, that L always contains an idempotent (we prove this in case L is the line or plane, or when M is two dimensional and L is a regular boundary). Clearly this would imply an affirmative answer to the question raised above for arbitrary dimensions. The existence of the open subgroup G gives us as corollaries two results of A. D. Wallace [1; 2], both of which are encompassed in the following more general result: if a compact manifold S with boundary B is a semigroup with identity, then the set of elements with inverses is contained in B or else is all of S. We consider also the following problem: if the closed right half plane is a semigroup in which the open half plane is a group G, how many possibilities are there for multiplication on the y-axis (=L)? We show that if G is isomorphic to the two dimensional vector group, there are exactly four possibilities, and examples are given of each of them. If G is the nonabelian group, we can identify two possibilities, but we are unable to prove these are all. (It might be interesting to note that even in the plane, we are unable to decide whether L need be a regular boundary for G or not.)

The recogllition of the abstract identity of geonletry and analysis, which results from the llotion of coordinates on the one hand and from the classical work of VON STAUDT t on the algebra of throws on the other, and which recent work oll the foundations of geometry has ftllly established, has brought with it a broader conception of the colltellt of geometry. It has meant llot only the introduction of imaginary elements alld the resulting conception of a complex space (of any number of dimensions), but it has also led to the consideration of geometries with respect to any number-system (finite or infinite), i. e., of spaces the elemellts of which may be determined by sets of numbers (coordinates) belonging to a givell number-system. t An important result of the recognition of the identity referred to is the etnphasis it places on the possibility of using geometric or synthetic methods in the solution of analytic problems. It seems likely that the fact that such methods have received comparatively little attention hitherto has resulted in a loss of power. The present paper, it is hoped, will telld to substantiate this assertion. We are llere concerned with certaill fundamental problems in the projective geotnetry on a comptex plane, i. e., a plane the points of which are determined by sets of llomogeneous coordinates (xl, X2, X3), where the xi are any ordinary complex numbers. not all zero. Though the illvestigation is essentially geometric alld the results are susceptible of immediate application to problems of importance in geoInetry, these results are of even greater interest in the theory of functions of two complex variables. To make this clear we shall glance briefly at the corresponding problems on a complex line, tlle results of which are well-known ill the theory of functions

Introduction. Crossed-inverse (C.I.) loops are loops in which the identity xy x'=y holds for all x, y of the loop, x' being the right inverse of x. Such loops have already been studied [1; 2], and the results were useful in the study of neofields [6] and of geometrical subjects [3]. C.I. loops are special automorphic-inverse loops, that is, loops satisfying the identity (xy)'=x'y', for all x, y in the loop. The first section of this paper deals with isotopic C.I. loops, and they are proved to be also isomorphic. Then the autotopisms of C.I. loops are studied, and the main result of this part is the fact that the "companions" of all the autotopisms, i.e. the elements that correspond to the translation elements of the isotopisms, form a subloop which is commutative and Moufang. ?2 of the paper gives a necessary condition for the existence of finite automorphicinverse loops consisting only of the unit eleinent and inverse-cycles [cf. 1 ] of the same length. The condition concerning the equal length of all inversecycles is necessary for the transitivity of the automorphism group of the loop [2] and hence for the possibility of considering the loop as the additive loop of a neoring. The third part deals with a method of constructing infinite automorphic-inverse loops satisfying also certain other identities. The author is grateful to R. H. Bruck for fruitful discussions on the subject of this paper. 1. 1. Definitions. 1.11. The biunique mapping which maps every element, c, of a loop (G, *) on its right inverse will be called J. Thus c cJ= 1, and cJ-1 c= 1. In automorphic-inverse loops J is an automorphism. 1.12. In the principal isotope of the loop G, with the new operation (*) such that ag*fb =ab, for all a, b in G, the ordered pair (g, f) is called the pair of translation elements of the principal isotope. 1.2. Elementary properties of C.I. loops. 1.21. The identities ab aJ=b and a(b aJ) =b are equivalent. Proof. If P and Q are biunique mappings of the loop onto itself, then PQ= I, the identity mapping, if and only if QP=I. Let L(x) and R(x) be

"associator dependent rings" and we shall adopt this convention [6]. It is immediate that the right and left alternative rings, the flexible rings, the rings of type (y, 6) and the anti-flexible rings all belong to the class of associator dependent rings. Linearizing (2) yields (1) with ci = 1 for i = 1, ., 6. Since (2) is certainly not strong enough to give a structure theory, we assume further that A satisfies an identity of the form (1) which is not implied by (2). We show in ?2 that A must then satisfy one of the following identities

The concept of a semnigroup satisfying an identity xy=f(x, y) is generalized by considering identities in n-variables and letting the identity depend on the variables. The property of satisfying a "variable identity" is studied. Semigroups satisfying certain types of identities are characterized in terms of unions and semilattices of groups. Introduction. Semigroups satisfying an identity of the form xy=f(x, y) have been studied by Tully [5] and Tamura [4]. In [2], we generalized Tamura's result. We considered semigroups S satisfying: for every a, b E S there exists a positive integer m such that ab=b"la"l .. b"ma"m where A,, [t are integers greater than one, i= 1, ..., m, and Em= 1 lt, > 2. We proved that a semigroup S satisfies this condition if and only if S is an inflation of a semilattice of periodic groups. The purpose of this article is to consider semigroups satisfying the analogous condition for n variables. 1. Preliminaries. Throughout S will denote a semigroup and E=E(S) the set of idempotents of S and n will be an integer, n> 2. Let Fn denote the free (noncommutative) semigroup generated by the distinct letters x1, .. ., x". Denote by Cn the subsemigroup of Fn consisting of all elements x E Fn each of which is the product of all of the x1, ... ., xn, allowing repeated use. Let Rn denote the semigroup ring of Cn over the integers Z. Thus Rn is the set of functions of finite support from Cn to Z. It is well known that Rn can be considered as the set of finite formal sums of elements in Cn and coefficients in Z. DEFINITION. We define a subset n of Rn by Y={f |fE Rn,f=fi-f2 there exists c1, c2 E Cn5, c1# :c2 such that f(c) = O unless c= cl, f(ci) = I }. Thus Y'I is the set of all fE Rn which are the differences of two different monomials. That is, fecY" if and only if f=Xk1 * Xkr-Xll . . Xs1#0 where each xi (i= 1, .. ., n) appears at least once in xk1 xc, and at least once in xll . xls. If a1, .. ., an E S there exists a homomorphism p: Cn -? S such that p(xi)= ai. This homomorphism extends to a homomorphism (p' from Rn into the semigroup ring of S over Z. If f=f1-f2 E Y' and a1, .. ., an E S we say that f1(a1, .. ., an) =f2(al ... ., an) if fe ker p'. Thus f1(al, ..., a.) =f2(a1, . .., an) if and only if considering f1 and f2 as monomials in x1, .. ., xn when a, is substituted for xi and Received by the editors April 19, 1971. AMS 1969 subject classifications. Primary 2093.

for fixed ai CF. II. There is an algebra B over F such that B satisfies (1), B has an identity element, and B is nonflexible; that is, there are elements x and y in B such that x(yx) # (xy)x. These conditions are similar to those used by Albert to define almost left alternative algebras [5 ]. Albert's paper led to the study of algebras of (y, 8) type by Kleinfeld and Kokoris [9; 1 1; 12 ]. Kokoris has shown that any simple finite dimensional algebra of characteristic prime to 30 of ('y, 8) type is either alternative or has an identity element which is an absolutely primitive idempotent(2). Our alteration of Albert's conditions yields a new class of simple power-associative algebras. We note that property II seems more natural in light of Oehmke's results [13] and the remark that most of the well-known nonassociative algebras (Jordan, noncommutative Jordan, Lie, alternative, associative) satisfy the flexible identity x(yx) = (xy)x. In ?2 it is shown that if F is algebraically closed then any algebra A over F belonging to W is quasi-equivalent in F to an algebra A (yA) where A (y) satisfies one of the following identities: (i) (xy)z-x(yz) = (zy)x-z(yx), (ii) x(xy) +(yx)x= 2(xy)x, (iii) x(xy) + (yx)x = (xy)x+x(yx), (iv) x(yz+zy) +(yz+zy)x=(xy+yx)z+z(xy+yx). The remainder of the paper is devoted to the study of algebras which satisfy some one of these four identities. We find that any power-associative

An n-braid is called k-decomposable if and only if the removal of k arbitrary strands results in a trivial (n k)-braid. k-decomposable n-linkages are similarly defined. All k-decomposable n-braids are generated by an explicit geometric process, and so are all k-decomposable n-linkages. The latter are not always closures of k-decomposable n-braids. Many examples are given. Introduction. This paper results from the investigation of linkages with the property that the removal of any arbitrary link results in the remaining links falling apart from each other. Such linkages have been termed "decomposable", and in the course of examining them, it has been necessary to weaken this property as follows. If the removal of any arbitrarily chosen k links from a linkage on n links results in the remaining n k links falling apart from each other, the original linkage on n links is termed "k-decomposable". ?I consists of a sequence of lemmas resulting in theorems giving various constructive enumerations of families of linkages, including a construction for the family of all decomposable linkages. ? II consists of examples of families of decomposable linkages of interest. I wish to thank Professor Wilhelm Magnus, who posed problems treated in this paper, and helped kindly and patiently in their solutions. I. The geometric construction of decomposable braids and linkages. Definition 1. A braid [21 on n strands shall be called decomposable iff whenever a single arbitrary strand is removed, the remaining braid on n 1 strands is deformable into the identity braid on n 1 strands; For any braid to be decomposable, it is necessary for each strand to end in the same position in which it began. We may therefore restrict our attention to the subgroup, In' of elements of the braid group, Bn, on n strands, which leave strand positions the same at their ends as at their beginnings. Presented to the Society, April 9, 1971; received by the editors November 8, 1971. AMS (MOS) subject classifications (1970). Primary 55A25. Key vords and phrases. Decomposable, braids, linkages, groups. ( 1) The results in this paper constituted part of the author's Ph.D. thesis written at the Courant Institute of Mathematical Sciences, New York University, under the direction of Professor Wilhelm Magnus. The author further wishes to acknowledge and thank the Rutgers University Research Council for having provided funds for the typing of the manuscript of this paper. Copyriglt O 1(73 Ainerican Mathematical Society 111 This content downloaded from 157.55.39.163 on Wed, 21 Sep 2016 05:25:15 UTC All use subject to http://about.jstor.org/terms

In [1] Clifford showed that the structure of any bisimple inverse semigroup with identity is uniquely determined by that of its right unit subsemigroup. The object of this paper is to show that the structure of any bisimple inverse semigroup with or without identity is determined by that of any of its a-classes. Let us define a right partial semigroup S to be a set S together with a partial binary operation satisfying the following condition: (A) if, for elements a, b, c of S, a(bc) is defined then so also is (ab)c defined and then a(bc) = (ab)c. We say that a right partial semigroup S is isomorphic with a right partial semigroup T if there exists a bijection b of S onto T such that ab is defined if and only if aq bq is defined and such that if ab is defined then (ab)k = aq bb. We define an RP-system (R, P) to be a right partial semigroup R together with a subsemigroup P of R such that: P(1) ab is defined if and only if a E P, for all a, b in R; P(2) R has a left identity contained in P; P(3) ac = bc implies that a = b for all a, b E P, c E R; P(4) foralla,be R,PanPb=Pcforsomece R. It then follows from P(1) and P(3) that any left identity of R contained in P is, in fact, a two-sided identity for P and so is unique. Now consider any a-class R of a bisimple inverse semigroup S. If we define the partial binary operation o in R by the rule that a o b = ab if and only if ab e R, then with respect to this operation R is a right partial semigroup with a subsemigroup P such that (R, P) is an RP-system (Theorem 1.4). Note 1. RP-systems can be obtained from systems having products more generally defined than stipulated in P(1). We then just ignore products which do not satisfy the condition P(1) (cf. Examples 1, 2 in ?6). Note 2. In particular, R could be a lattice ordered group and P the positive cone of R. We show that for any RP-system (R, P) there exists a bisimple inverse semigroup, which we denote by R-1 o R, some a-class of which is isomorphic with R. Conversely, for any a-class R of a bisimple inverse semigroup S there is a subsemigroup P of R such that (R, P) is an RP-system and R1 o R is isomorphic with S.

When one studies periodic transformations such as, for example, rotations, he often encounters transformations which are not periodic but which are, in a very real and non-technical sense, almost periodic. For instance, repeated rotation through an angle which is an irrational part of a revolution will never bring a point set back point-for-point into itself; yet this object may be as nearly attained as we please by repeating the process an appropriate number of times. Moreover, such "appropriate" numbers are relatively denset among the integers. This example suggests the definition of an a.p. (almost periodic) transformation; it being only necessary to make precise the meaning of "as nearly as possible" when applied to bringing the points of a set back into themselves. Consider, for example, a set Z of uniformly continuous transformations which take each point of a complete metric space G into a point of G. Let Z contain the identity and the product of any two of its elements. Then if t is a variable point of ( and 4(a) and T(t) are any two elements of Z, let the smaller of the two quantities, unity and the least upper bound for all t in ( of the distance between the points b(a) and T'(), be called the distance between the transformations 4(a) and T'(), and let it be indicated by I| 4(a), IT (t) I. Then || 4(e), t || represents one way of telling how nearly the points 4i(t) approximate the points t. Moreover a transformation of Z will be called a.p. if to each positive number e there corresponds a positive integer L so great that among each L successive positive integers there is an integer N satisfying || 4 N(t) I | I-e. This is merely an example of a definition of an a.p. transformation. A more general definition will be given in the next section. Transformations will be thought of as points in a new space, and a.p. points will be defined. Moreover, for simplicity in wording and notation, most of the theorems on a.p. transformations will be stated in terms of a.p. points. However, the reader may readily re-phrase them in terms of the more natural and significant a.p. transformations.