We construct all semirings with a given completely simple additive semiroup ℳ(G, I, ∧, P) by means of an associative and distributive multiplication on G, and associative multiplications on I and ∧ satisfying certain conditions.

The tensor product G ⊕ H of graphs G and H is the graph with point set V(G) × V(H) where (υ1, ν1) adj (υ2, ν2) if, and only if, u1 adj υ2 and ν1 adj ν2. We obtain a characterization of graphs of the form G ⊕ H where G or H is K2.

Banaschewski (1963) and Frink (1964) generalized the compactification procedure of Wallman to obtain Hausdorff compactifications of Tychonoff spaces. Numerous papers have been devoted to the problem whether all Hausdorff compactifications may be obtained in this way, and for many classes of compactifications an affirmative answer has been given. This note is a contribution in this direction. We show that if a (Hausdorff) compactification αX of X is the quotient space of a Wallman compactification γX in such a way that the set of multiple points of αX with respect to γX is not too large, then αX too is a Wallman compactification. The results are generalizations of earlier results of Steiner and Steiner (1968) and by the author (1966) for the special case that γX is the Stone Čech-compactification.

Let G be an abelian lattice ordered group (an l-group). If G is, in fact, totally ordered, we say that G is an 0–group. A subgroup and a sublattice of G is an l-subgroup. A subgroup C of G is called convex if 0 ≦ g ≦ c ∈ C and g ∈ G imply g ∈ C, C is an l-ideal if C is a convex l-subgroup of G. If C is an l-ideal of G, then G/C is also an l-group under the canonical ordering inherited from G. If, in fact, G/C is an 0–group, then C is said to be a prime subgroup of G.

Two generalized mean value theorems, for functions with values in a linear locally convex topological space, are proved, as consequences of two theorems for real valued functions of real variable.

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

Let R be a ring (with identity) and P(R) denote its prime radical. R is called semi-prime when P(R) = (0). If G is a group, the group ring of G over R will be denoted by RG.

A tight Riesz group G is a partially ordered group G that satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in (1970) and the tight interpolation property has been considered by Fuchs (1965), Miller (1973), (to appear), (preprint), Loy and Miller (1972) and Wirth (1973). If G is free of elements called pseudozeros then G is a non-discrete Hausdorff topological group with respect to the open interval topology U. Moreover the closure P of the cone P of the given order is the cone of an associated order on G. This allows an interesting interplay between the associated order, the tight Riesz order and the topology U. Loy and Miller found of particular interest the case in which the associated partial order is a lattice order. This situation was considered in reverse by Reilly (1973) and Wirth (1973), who investigated the circumstances under which a lattice ordered group, and indeed a partially ordered group, permits the existence of a tight Riesz order for which the initial order is the associated order. These tight Riesz orders were then called compatible tight Riesz orders. In Section one we relate these ideas to the topologies denned on partially ordered groups by means of topological identities, as described by Banaschewski (1957), and show that the topologies obtained from topological identities are precisely the open interval topologies from compatible tight Riesz orders.

In this paper we extend a method of Ehrenfeucht and Fraissé to second-order theories and use this extension to prove the decidability of the second-order theory of an equivalence relation.

Throughout this paper only abelian l-groups will be considered and G will denote an abelian l-group. G is large in the l-group H or H is an essential extension of G if G is an l-subgroup of H and for each l-ideal L ≠ 0 of H we have L ∩ G ≠ 0. A ν-hull of G is a minimal vector lattice that contains G and is an essential extension of G. Each G admits a ν-hull (Conrad (1970)).We shall be interested in the following properties of G.

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

Rudin's synthesis method for investigating closed subalgebras of L1(G), where G is an infinite compact abelian group, is extended to the study of closed subalgebras in homogeneous Banach algebras and Segal algebras. Necessary and sufficient conditions are given for the synthesis to hold in certain classes of homogeneous Banach algebras and it is proved that in the Ap(G) algebras the synthesis holds for 1 ≦ p 2 but fails for Ap(T), 2 < p < ∞.

We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.