We give a complete axiomatization for the modal μ-calculus on finite trees. While the completeness of our axiomatization already follows from a more powerful result by Igor Walukiewicz in [11], our proof is easier and uses very different tools, inspired from model theory. We show that our approach generalizes to certain axiomatic extensions, and to the extension of the μ-calculus with graded mo...

If φ is a scattered order type, μ a cardinal, then there exists a scattered order type ψ such that ψ → [φ] μ,א0 holds. In this note we prove a Ramsey type statement on scattered order types. A trivial fact on ordinals implies the following statement. If μ is an infinite cardinal, then μ → (μ+)1μ. It is less trivial but still easy to show that if φ is an order type, μ a cardinal then there is so...

Jachymski [ Proc. Amer. Math. Soc., 136 (2008), 1359-1373] gave modified version of a Banach fixed point theorem on a metric space endowed with a graph. In the present paper, (G, Φ)-graphic contractions have been de ned by using a comparison function and studied the existence of fixed points. Also, Hardy-Rogers G-contraction have been introduced and some fixed point theorems have been proved. S...

Let C be a unital AH-algebra and let A be a unital separable simple C-algebra with tracial rank no more than one. Suppose that φ, ψ : C → A are two unital monomorphisms. With some restriction on C, we show that φ and ψ are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A) τ ◦ φ = τ ◦ ψ for all tracial states of A and φ = ψ, where φ and ψ are homomorphisms from U0(C)/CU(C) →...

Our main goal is to give proofs of all results announced by Oleg Izhboldin in [13]. In particular, we establish Izhboldin’s criterion for stable equivalence of 9-dimensional forms. Several other related results, some of them due to the author, are also included. All the fields we work with are those of characteristic different from 2. In these notes we consider the following problem: for a give...

The notion of logic adopted by Abstract Algebraic Logic stems from Tarki’s. By a logic we understand any consequence relation on the algebra of formulas of some propositional language, namely any relation ` between sets of formulas and formulas such that 1. φ ` φ; 2. if Γ ` φ and Γ ⊆ ∆, then ∆ ` φ; 3. if for every ψ ∈ Γ , ∆ ` ψ and Γ ` φ, then ∆ ` ψ; 4. if Γ ` φ, then for any substitution σ of ...

A formal theory of information should represent A consistent system of questions Aggregation or combination of information Focussing or Extraction of information Information algebra (Φ, D) cover these aspects: D a lattice of domains (questions) Φ a semigroup under combination ∀x ∈ D, φ ∈ Φ, φ ⇒x is the information in φ relating to x. Axiomatics Formal definitions and axioms: (Φ, D) is a two-sor...

A1 Either a property or its negation is positive, but not both: ∀φ[P (¬φ)↔ ¬P (φ)] A2 A property necessarily implied by a positive property is positive: ∀φ∀ψ[(P (φ) ∧ ∀x[φ(x)→ ψ(x)])→ P (ψ)] T1 Positive properties are possibly exemplified: ∀φ[P (φ)→ ♦∃xφ(x)] D1 A God-like being possesses all positive properties: G(x)↔ ∀φ[P (φ)→ φ(x)] A3 The property of being God-like is positive: P (G) C Possib...