A monodromy theorem for homomorphisms of local groups into groups is proved. It follows that under suitable conditions the universal group of the local group depends only on the germ of the local group (up to natural isomorphism).

1980 Mathematics subject classification (Amer. Math. Soc.): 22 E 05.

An extension of the Banach-Mackey theorem is used to prove a theorem about countable families of closed balanced convex sets that cover a product of linear topological spaces. This theorem clarifies proofs that certain Baire-type properties, including the unordered Baire-like property, are preserved under products. A modification of the theorem is used to show that a property involving the bounded-absorbing sequences of DeWilde and Houet is also productive. Finally, a question is posed about balanced absorbing sets relating to products of linear Baire spaces.

1980 Mathematics subject classification (Amer. Math. Soc.): 46 A 99.

Let I be a set and let (I) denote the set consisting of the 0 matrix over I × I and the matrix units over I × I. Then for x, z in (I) and x≠0≠z, xyz≠0 has precisely one solution y. This and several other statements are shown to be equivalent characterizations of (I) regarded as a semigroup with zero.

1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

If G and H are infinite groups then G is said to be larger than H (H≼G) if there are subgroups A of G, B of H, each of finite index, such that B is an epimorphic image of A. Pride (1979) showed that if G has finite ‘height’ with respect to the quasi-order ≼ then there are only finitely many (classes of) minimal groups H with H ≼G, and asked whether this were true without the minimality restriction on H. This paper gives a negative answer to his question by exhibiting a group G of height four with infinitely many (classes of) groups H satisfying H≼G.

1980 Mathematics subject classification (Amer. Math. Soc.): 20 E 99, 20 K 15.

Certain semigroups of continuous selfmaps of the closed unit interval are shown to have the property that all their automorphisms are inner. Contrary to expectation, certain other such semigroups do have outer automorphisms.

1980 Mathematics subject classification (Amer. Math. Soc.): primary 20 M 20; secondary 54 C 40.

Every τ-smooth Borel measure is support-concentrated. We shall prove in this note that the converse of this statement is not true, in general. Furthermore, we shall give some conditions assuring that a support-concentrated Borel measure be τ-smooth.

1980 Mathematics subject classification (Amer. Math. Soc.): 28 C 15.

A composite integer N is said to be a strong pseudoprime for the base C if with N – 1 = 2sd, (2, d) = 1 either Cd = 1, or C2r ≡ 1 (mod N) some r, 0 ≤ r < s. It is shown that every arithmetic progression ax+b (x = 0,1, …) where a, b are relatively prime integers contains an infinite number of odd strong pseudoprimes for each base C ≤ 2.

1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 15.

In a recent paper of the author the well-known Vagner-Preston Theorem on inverse semigroups was generalized to include a wider class of semigroups, namely right normal right inverse semigroups. In an attempt to generalize the theorem to include all right inverse semigroups, the notion of μ – μi transformations is introduced in the present paper. It is possible to construct a right inverse band BM(X) of μ – μi transformations. From this a set AM(X) for which left and right units are in BM(X) and satisfying certain conditions is constructed. The semigroup AM(X) so constructed is a right inverse semigroup. Conversely every right inverse semigroup can be isomorphically embedded in a right inverse semigroup constructed in this way.

1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 20.

The interrelationships between norm convergence and two forms of convergence defined in terms of order, namely order and relative uniform convergence are considered. The implications between conditions such as uniform convexity, uniform strictness, uniform monotonicity and others are proved. In particular it is shown that a σ-order continuous, σ-order complete Banach lattice is order continuous.

1980 Mathematics subject classification (Amer. Math. Soc.): 46 A 40.

A distribution function (F on [0,∞) belongs to the subexponential class if and only if 1−F(2) (x) ~ 2(1−F(x)), as x→ ∞. For an important class of distribution functions, a simple, necessary and sufficient condition for membership of is given. A comparison theorem for membership of and also some closure properties of are obtained.

1980 Mathematics subject classification (Amer. Math. Soe.): primary 60 E 05; secondary 60 J 80.

In this work we demonstrate that if Ω ⊂ Rn (n ≧ 3) is either a half space or a unit ball, and if E ⊂ ω then E is an ordinary thin set at a boundary point of Ω (including the point at infinity if Ω is a half space) if and only if it is a full-thin set at the corresponding Kuramochi boundary point of Ω. The case for n = 2 has already been considered in an earlier work.

1980 Mathematics subject classification (Amer. Math. Soc.): 31 B 05.

Let p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.

1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

Let N be a positive integer. This paper is concerned with obtaining bounds for (p prime), when N is an odd perfect number, a multiperfect number, or a quasiperfect number, under assumptions on the existence of such numbers (where none is known) and whether 3 and 5 are divisors. We argue that our new lower bounds in the case of odd perfect numbers are not likely to be significantly improved further. Triperfect numbers are investigated in some detail, and it is shown that an odd triperfect number must have at least nine distinct prime factors.

1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 20.