Let B be a complex unital Banach algebra. We consider the Banach algebra A = B × C ordered by the algebra cone K = {(a, ξ) ∈ A : ‖a‖ ≤ ξ}, and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in B. 1. Ordered Banach algebras The aim of this paper is to stress the aspect of the applicability of the ordered Banach algebra...

have a solution in positive integers (x, y, z) other than that given by x = 5, y = 7 and z = 4. A negative answer to this question follows from a classical result of Ljunggren [8], as recently refined by Cohn [4]: Theorem 1.1. Let the fundamental solution of the equation v2−Du2 = 1 be a + b √ D (i.e. (v, u) = (a, b) is the smallest positive solution). Then the only possible solutions of the equ...

This paper discusses the notion of projectivity and algorithms for projectivizing and detecting non-projective edges in totally ordered rooted trees (such trees are used in dependency syntax analysis of natural language, where they are called dependency trees). In the first part, we review the notion of projectivity, then we present a new definition inspired by the algorithmic inquiry and show ...

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ≤ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other thing, we prove that e...

Sorting of transitive total ordered sets is one of the best investigated topics in Computer Science. A generalization of this is sorting an intransitive total ordered set which is more complex. While a transitive total ordered set (e. g. a subset of the real numbers) holds a ≥ b∧b ≥ a⇒ a ≥ c, this does not have to be true for a intransitive total ordered set. Only a → b ∨̇ b → a is claimed. Neve...

Given a mapping f : A → B from a (pre-)ordered set A into an unstructured set B, we study the problem of defining a suitable (pre-)ordering relation on B such that there exists a mapping g : B → A such that the pair of mappings (f, g) forms an adjunction between (pre-)ordered sets. The necessary and sufficient conditions obtained are then expressed in terms of closure operators and closure syst...

Let (A,1) be a partially ordered set. The relation a 1 b can be read as “a precedes b”. For elements a and b in A, we write a ≺ b if and only if a 1 b and a 6= b holds. For notational convenience, we also write a o b if and only if a 1 b holds, and a  b if and only if a ≺ b holds. We call (A,1) a well-founded set if and only if every non-empty subset M of A contains at least one minimal elemen...

We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets represents \geometries" which are more general than projective planes. It was shown in 2], 4] and 6] that...

In this paper, we introduce the notion of an extended metric space ($p$-metric space) as a new generalization of the concept of $b$-metric space. Also, we present the concept of $(psi ,varphi )_{Omega}$-contractive mappings and we establish some fixed point results for this class of mappings in ordered complete $p$-metric spaces. Our results generalize several well...