Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

In an abelian category, a commutative quadrangle is called bicartesian if its diagonal sequence is short exact, i.e. if it is a pullback and a pushout. A commutative quadrangle is bicartesian if and only if we get induced isomophisms on the horizontal kernels and on the horizontal cokernels. In a triangulated category in the sense of Verdier [3, Def. 1-1], a commutative quadrangle is called hom...

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

An analogue of the Kunz-Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings. Let S be a commutative semigroup (we always assume that S contains a neutral element), and K a field. For every m 6 Z+ the assignment x H-» x, x £ S, induces a K-endomorphism 7m of the semigroup ring R = K[S]. Therefore we can consid...

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true”. The concept of Hilbert algebras was introduced in the 50-ties by L. Henkin and T. Skolem (under the name implicative models) for inve...

The generalized state space of a commutative C-algebra, denoted SH(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C-extreme point of SH(C(X)) satisfies a certain spectral condition on...