Given two topologies J1, J2 on a set X, J1 is said to be coarser than J2, written J1 ≦ J2, if every set open under J1 is open under J2. A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.
w 1. The growth seriesAs originally conceived (see for example Milnor [8], Bass [2]), a growth series was associated with a finitely generated group G and a set of generators {gt ..... gn}: the n-th term a, in the growth series is then the number of elements of the group G which can be written as a word of length n but no shorter in the given generators and their inverses. Most of the interest in the literature centres on the rate of growth of the a,, rather than on the growth function, the function whose power series is ~ a, x".It is natural to consider non-symmetric generating sets (that is, sets which generate the group as a semigroup, but which do not necessarily contain inverses); and, particularly if subgroups are to be included in a general theory, to allow the generators to be considered of length other than 1 (see for example Cannon [5], Wagreich [11], Benson [4]). Thus the natural setting for the concept appears to be that of filtered semigroups. Definition. A filtered set is a set X together with a collection of subsets X s for non-negative integers s such that Xs~_X~+ t and X= ~ Xs; the elements of X s -X~_ 1 are said to be of degree s.A filtered semigroup is a semigroup G whose underlying set is filtered in such a way that G0={identity } and Gs. Gt~G~+ t. Throughout this work we shall further insist that G~ is a finite set for each s.The growth function 7~(x) of a filtered semigroup is the power series ~ a~x ~ where a~=card(Gs-G~ 1). As has been noted by others (see Wagreich [11]), the growth function can be identified with the Hilbert-Poincare series of a filtered algebra, or more precisely, the associated graded algebra. Choose a co-efficient ring R: in this paper R will be either the ring of integers or a field. The algebra RG inherits a filtration from G, and a t is the R-rank of (RG)s/(RG)s_ 1. (Note that our filtrations are always increasing, and for a filtered R-algebra F we assume F o =R.) The question of the rationality of this series in the setting of more general graded algebras is of current interest: see for example [1, 6].
In this note the row and column ideal class invariants of a matrix (cf. [3]) are applied to the Alexander matrix of a knot to give an invariant for knots. We give an example of the use of this invariant to distinguish a pair of knots which cannot be distinguished by their elementary ideals, torsion numbers, or linking invariants.1 1. Let M be an mXn matrix over a commutative integral domain R. For any integer k consider the &th compound matrix Mik), the GK) matrix whose entries are the kXk minor determinants of M, rows and columns written in lexicographic order. If the rank of M is r, Mm = 0 for k greater than r, whereas M(r) contains some nonzero element. Further, M is equivalent to a diagonal matrix N over the quotient field of R and, as may be easily seen, Mlk) is equivalent to Nm. Since the rank of A(r) is 1, the rank of Af(r) is 1.We recall that two ideals /, J are said to be equivalent if there exist nonzero principal ideals P, Q such that PI = QJ. The ideal classes of a ring under this relation form a semi-group with the principal ideal class as identity.By the ith row ideal p¿ of M is meant the ideal generated by the elements of the ith row of MM. Suppose that both the ith and &th rows contain nonzero elements. Writing M(r) = ||my|| and choosing q such that JM.j is nonzero, we have for any j miq ma det =0 mkq mkj since Mw is of rank 1. Thus miimkj = mkqmij and (mit)pk = (mkq)pi, where (z) denotes the principal ideal generated by z. It follows that mkq is nonzero and thus that any two nonzero row ideals are equivalent. This equivalence class of ideals is called the row class of M. Similarly we may define the column class of M.2. There is defined for matrices over a ring R an equivalence relation by means of the following operations (see [l, p. 101]) ;
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