In case K is a division ring, the set S coincides with the set generated by the perfect squares as used by Szele. This follows easily from the identity xyx = (xy)2(y~1)2y, x, yEK*. If the domain of integrity K is ordered, say K* = P\J( — P), then for an extension L of K the ordering of K can be extended to an ordering of L if and only if TEL*, where T is the additive semigroup in L generated by...

having all the rules of Hilbert’s positive logic (cf. for example [3, p. 98]), the connective of negation ¬ is said to be conditional iff ` ⊆ |=M, where |=M is the consequence relation determined on L by a model M =< I,≤, φ, A > (with a nonempty partially ordered set < I,≤>, a valuation φ ⊆ I × V , V is the set of propositional variables, and A ⊆ I, ∅ 6= A 6= I) in the standard way: for any X ⊆...

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields. 1. Existentially closed real-closed difference fields. In the first part of this paper we will consider on one hand difference totally ordered fields, namely totally ordered fields with a distinguished automorphism σ and on the other ...

Contents 1. Introduction. 2. Heirs. 3. The invariance group of a cut. 4. Review of T-convex valuation rings. 5. The invariance valuation ring of a cut. 6. A method for producing cuts with prescribed signature. 7. Existentially closed extensions. 1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (p L , p R) of subsets p L , p R of M such that p L ∪ p R = M and ...

The paper [3] started a new approach to Abstract Algebraic Logic in which, instead of the usual equivalence-based classification of logical systems leading to the well-known Leibniz hierarchy of protoalgebraic logics (see [5]), we proposed an alternative setting based on implication connectives. By studying the properties of weak p-implications (understood as generalized connectives defined by ...

In this paper, we tackle the study of so-called supplementarity measures. They are introduced axiomatically and are then related to incompatibility measures by antonyms. To do this, we have to establish what we mean by antonymous measure. We then prove that, under certain conditions, supplementarity and incompatibility measures are antonymous. Besides, with the aim of constructing antonymous me...

ABSTRACT. Let (a, 6) be any open subinterval of the reals which contains the origin and let 33 denote the family of all distributions on [a, b) which are regular in some interval (e, 0), where e < 0. Then S is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on (a, b) whose supports are contained in [0, b), it is ordinary convolution. Also, S) can...

Let $x$ and $y$ be two distinct vertices in a connected graph $G$. The $x,y$-location of a vertex $w$ is the ordered pair of distances from $w$ to $x$ and $y$, that is, the ordered pair $(d(x,w), d(y,w))$. A set of vertices $W$ in $G$ is $x,y$-located if any two vertices in $W$ have distinct $x,y$-location.A set $W$ of vertices in $G$ is 2-located if it is $x,y$-located, for some distinct...