Let P be a partially-ordered set in which every two elements have a common lower bound. It is proved that there exists a lower semilattice L whose elements are labelled with elements of P in such a way that (i) comparable elements of L are labelled with elements of P in the same strict order relation; (ii) each element of P is used as a label and every two comparable elements of P are labels of comparable elements of L; (iii) for any two elements of L with the same label, there is a label-preserving isomorphism between the corresponding principal ideals. Such a structure is called a full, uniform P-labelled semilattice.

Let V be a variety of groups in which the free group is residually finite, and let S be a residually finite semigroup. Let Nv(S) be the ‘free’ distributively generated near-ring constructed from S and V. Theorem; Nv(S) is residually finite.

If w is a group word in n variables, x1,…,xn, then R. Horowitz has proved that under an arbitrary mapping of these variables into a two-dimensional special linear group, the trace of the image of w can be expressed as a polynomial with integer coefficients in traces of the images of 2n−1 products of the form xσ1xσ2…xσm 1 ≤ σ1 < σ2 <… <σm ≤ n. A refinement of this result is proved which shows that such trace polynomials fall into 2n classes corresponding to a division of n-variable words into 2n classes. There is also a discussion of conditions which two words must satisfy if their images have the same trace for any mapping of their variables into a two-dimensional special linear group over a ring of characteristic zero.

A class of Markov chains is considered for which a certain property of the tail events makes bounded harmonic functions obtainable from bounded space-time harmonic functions. Applications to almost surely convergent Markov chains are given and, in particular, a representation of Martin-Doob-Hunt type is derived for all bounded harmonic functions of a finite mean supercritical branching process.

Let R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

It is shown, using the author's construction for ‘labelled semilattices’, that every partially ordered set, in which every two elements have a common lower bound, is isomorphic to the partiallyordered set of 𝒥-classes of some completely semi-simple inverse semigroup.

1980 Mathematics subject classification (Amer. Math. Soc): primary 20 M 10, secondary 04 A 05, 08 A 05.

N(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

Let 〈fn≧0 be nonnegative real numbers with generating function f(x) = Σfnxn. Assume f(x) has the following properties: it has a finite nonzero radius of convergence x0 with its only singularity on the circle of convergence at x = x0 and f(x0) converges to y0; y = f(x) satisfies an analytic identity F(x, y) = 0 near (x0, y0); Fy(l) (x0, y0)= 0, 0 ≦ i < k and Fy(k) (x0, y0) ≠ 0. There are constants γ, a positive rational, and c such that fn~cx0−n n−(1 +ggr;). Furthermore, we show (i) in all cases how to determine γ and c from f(x) and (ii) in certain cases how to determine them from F(x, y).

The completions of certain nilpotent groups with respect to some ascending sequences of integral domains are constructed. These completions are generalizations of Lazard completions for the groups under consideration and they are Lie algebras over the first integral domain in the sequence. The construction is possible in particular for finite p-groups of exponent p and class < p.

It is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.

We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

A recent result of Rogozin on the relative stability of a distribution function is extended, by giving equivalences for relative stability in terms of truncated moments of the distribution and in terms of the real and imaginary parts of the characteristic function. As an application, the known results on centering distributions in the domain of attraction of a stable law are extended to the case of stochastically compact distributions.

I am grateful to W. Taylor for pointing out that Lemma 2.3 of Neumann (1978), which claimed that a variety and its finitary reduct have the same finite algebras, is incorrect. This weakens several results of the paper: the lattice homomorphism of Lemma 3.1 is not complete, Theorem 4.1 is false, and the Mal'cev conditions claimed to be strong in 6.1, 6.2 and 7.1 are not strong. Taylor (1980) shows one cannot improve on this.