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1U - 16 May, 19^9 ABSTRACTS OF CONTRIBUTED PAPERS The Fourth New Zealand Mathematics Colloquium held at the University of Otago, Dunedin, 1U - 16 May, 19^9 LOGIC AND FOUNDATIONS K. ASHTON, University of Auckland. Algebraic Structures in Linguistic Theory. Let A be a finite set and F(A) the free semigroup of strings over A. L is a "language" if L C F(A). An environment is a string of the form a *a , where a , a € F(A) and 3 x € A such that the string 12’ 1 2 a x a € L. Let E be the set of environments of L and denote a x a 1 2 1 2 by a 1*a2(x). Define E(x) = (e € EJe(x) € L) and W(e) = {x € Ate(x) € L} The binary relation x € W(e) (or e € E(x)) defines a polarity between A and E. Consequently, if for X C A we define E£(X) = ^X£x®(x ) ant* for C C E , VJC) = U w(c) it is well known that the mapping X -* W (E (X)) w. 2 CcC 2 2 K(X) is a closure operator on A. Further if E (X) = f ^ ^ ^ x ) an<3 W (C) = ^c£Cw(c ) then X "* W (Ex(X)) = H(X) is again a closure operator on A. If we interpret A as the vocabulary and L as the grammatically correct sentences of a language these closed sets, especially those closed under H,bear a close relationship to the concept of "grammatical category". Using the ordering x < y iff E(y) C E(x), we define a lattice on A in which the H-closed sets appear as principle ideals. Applying the same theory to the derivative concept of "string descrip­ tions" we obtain, as the suprema of principle ideals, a set of basic "phrase categories" in terms of which the lattice of "sentential forms" can be factorized. The (semantic) space C of concepts forms a lattice with principle ideals generated by "semantic categories" in terms of which the lattice of relations over C can be factorized. This latter space is also factorized as an image space of "sentential space". The factorization theorem for lattices now ensures a relation­ ship between any analysis of a language made on a purely syntactic (structural) basis and one making use of semantic criteria. References: 1. J. Kunze, Versuch eines objektivierten Grammatikmodells, Z. f. Phonetik Sprachwlss.Kommunikat.20(1967), M 5-UU8. 2. H.H* P6B8XH, IIpXKeHMe QOHSTHfl n9J6KeBTapHOft rpaMUaTH^eCKOj KaTeropHi". PI. Teuei CoBenaHna no MaTenara^ecKOft - jhhpBiCTKKe. le E M M V W ^ J959-, 5« S. Marcus, Sur un modele logique de la categorie grammaticsle elementaire.I, Rev. Math. Pures Appl. 1(1962), 91-107. W.R. BELDING, Victoria University of Wellington. Intuitionist Logic and Mathematics. A brief account will be given of the Intuitionist conception of mathematics as developed by Brouwer and Heyting. The relationship Intuitionist mathematics has to logic and language will be mentioned, showing why the Intuitionists reject the universal validity of the L E M (p V ~ p) and other theses of classical logic (e.g.~~p -► p). Hence non-constructive proofs are not universally acceptable in Intuitionist mathematics. Sufficient Intuitionist arithmetic and set theory is introduced to facilitate a proof of the Fan Theorem. It is argued that the Fan Theorem needs modification to: "If S is any fan and f an integer valued function defined on S such that f(s) is computed on the basis of a finite number of components ng of s (any element of S) then a finite maximum can be computed for the species of such ns". The Fan Theorem is the key theorem for Intuitionist analysis. ALGEBRA S.J. EERNAU, University of Otago. Bootstrap Constructions. The sort of construction involved in producing, say, the real numbers out of the rationals can be described as dragging out the extra structure by the bootstraps. All that is involved is the con­ sideration of appropriate subsets of the rationals (cuts or Cauchy sequences) and an equivalence relation. This basically simple idea recurs time and again, algebraic closure of a field, topological completion of a uniform space, Dedekind-MacNeille completion of a partially ordered set, quotient field of an integral domain. In my talk I shall outline two new situations in which the same process applies: the lateral completion of a lattice group and the free lattice hull of a partially ordered group. M. GRIFFIN, Applied Mathematics Division, D.S.I.R. Priifer Rings. Valuation theory provides a tool for studying the multiplicative ideal theory of Integral Domains. It is possible to generalize much of this theory to commutative rings with unit, where it can be used to study the regular ideals. Krull*s theorem that any integrally closed domain is an intersection of valuation domains may be general- ized, provided that the total quotient ring of the integrally closed ring involved is either einarting (all prime ideals are maximal) or has only a finite number of maximal prime ideals. There are fifteen or More conditions on an integral domain vhich are equivalent to being a Priifer domain. Most of these conditions generalize in such a way as to be equivalent and define a Priifer ring. However the natural gener­ alization of some conditions lead to Arithmetical rings and to commut­ ative semi-hereditary rings. Both of these families of rings may be obtained by placing special conditions on the total quotient ring of the Priifer rings. P. LORIMER, University of Auckland. The Construction of Projective Planes. Let F be a field of characteristic not 2 or 3, and let x3 - ax - 3 be a cubic polynomial irreducible over F. If a, b, c G F, let A . be the matrix a ° a ab-ipc -iPb-a^ The set of such matrices has the following two properties: (1) It is a vector space over F. (2) Except when a = b = c = 0, det Aafec ^ 0. These results can be used to construct a semi-field and hence a pro­ jective plane. If F is finite of order q, then the plane has order q3. In general, a semi-field can be constructed if we can find a set of n x n matrices over a field F which is a vector space of dimension n over F and, except for the zero matrix, consists of non-singular matrices. P.F. RENAUD, University of Canterbury. Geometry in Wormed Algebras. A geometrical property of a normed algebra refers to a property of th?> norm function itself rather than to one of the topology arising from this norm. For example, the existence and character of extreme points, vertices,etc. could be termed a geometrical property as these points would vary if a different, but equivalent, norm was chosen. If an algebra has an identity element e then the number of hyperplanes of support to the unit ball at e will depend upon the norm itself. At one extreme we may consider algebras which have only one such hyper­ plane. Recently Strzelecki has shown that if a real alternative algebra has this property then it must be one of the four classical algebras. This result is derived but Strzelecki's geometrical proof is replaced by a shorter analytical one. At the other extreme are algebras where the identity element is a vertex in the sense that the intersection of all hyperplanes of support is precisely the identity. It is shown that algebras possessing the "vertex property" lie somewhere between 4? strictly real algebras and algebras of complex type. Hie notion of a vertex leads naturally to a partial-order on the algebra. This order­ ing which generalizes a large number of orderings in particular cases would seem to merit further study. W.J. HONG, University of Notre Dame. Recent Work on Finite Simple Groups. A group G is simple if G is not the identity group (1} and G has no normal subgroup other than G and (1}. Since every finite group can in a certain sense be decomposed into a number of simple groups, a central problem of finite group theory is the determination of all the finite simple groups. The difficulty of this problem may be appreci­ ated by considering the great variety of known 6imple groups and the complexity of their structures, nevertheless, major advances have been made in the past decade. I. Classification theorems Several theorems have been proved, determining all the finite simple groups which satisfy some general condition. An example is the result of Feit and Thompson (19^3) that all simple groups of odd order are Abelian. II. Characterization theorems If G is one of tne known simple groups, a characterization of G is a theorem asserting that to within isomorphism G is the only finite simple group satisfying certain conditions. Such theorems have been proved for many of the known simple groups, for example the alternating groups (Kondo, 1968). III. Discovery of new simple groups A number of new simple groups have been discovered as a byproduct of the above work, and also in the study of a certain type of permut­ ation group. Host recently, several new groups have arisen in connect­ ion with a packing of spheres in 2^-dimensional Euclidean space. It remains to be seen if all these new groups can be fitted into some systematic framework. (For full text, see pp. 5-12) ANALYSIS D.P. ALCORN, University of Auckland. The Spectra of Tensor Prodoets of Operators. If A and B are bounded linear operators on a Hilbert space H there is a natural way of defining t'n> operator A (5) B on the tensor product space H ^5) H.
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