Volume 58, Issue 2

We will show that the bricks (of Bosbach) and the pseudo MV-algebras are each term equivalent to the class of semigroups with a pair of unary operations ^ and ˘ satisfying the equations: (âa)^b = b = b(aă)˘ and a($$ \hat b $$ a)˘ = (bă)^b and also show that a brick is an interval [0, u] of the positive cone of a unital lattice ordered group. We further extend the notion of implications to a pseudo MV-algebra and study the algebra of such implications.

For an MV-algebra let J 0() be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J 0() ≅ J 0(), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton.

In the paper continuous variants of densities of sets of positive integers are introduced, some of their properties are studied and formula for their calculation is proved.

Let K/Q be a cyclic tamely ramified extension of degree 6, then any ambiguous ideal of K has a normal basis.

Let (R,m) = k[x 1,..., x n](x 1,...,x n) be a local polynomial ring (k being an algebraically closed field), and Q:= (F 1,..., F r)R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e 0 (QR/(F 1)R, R/(F 1)R).

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

We prove an Alexandroff decomposition type theorem, which extends a decomposition theorem proved in [de LUCIA, P.—MORALES, P.: Decomposition of group-valued measures in orthoalgebras, Fund. Math. 158 (1998), 109–124].

In this paper, we are concerned with the oscillation properties of the third order differential equation $$ \left( {b(t) \left( {[a(t)x'(t)'} \right)^\gamma } \right)^\prime + q(t)x^\gamma (t) = 0, \gamma > 0 $$. Some new sufficient conditions which insure that every solution oscillates or converges to zero are established. The obtained results extend the results known in the literature for γ = 1. Some examples are considered to illustrate our main results.

Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral difference equations of the form $$ \Delta ^2 (r(n)\Delta ^2 (y(n) + p(n)y(n - m))) + q(n)G(y(n - k)) = 0 $$ and $$ (NH) \Delta ^2 (r(n)\Delta ^2 (y(n) + p(n)y(n - m))) + q(n)G(y(n - k)) = f(n) $$ are studied under the assumption $$ \sum\limits_{n = 0}^\infty {\tfrac{n} {{r(n)}} = \infty } $$, for various ranges of p(n). Sufficient conditions are obtained for the existence of bounded positive solutions of (NH).

If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive. It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable. To a quadratic polynomial x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.