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1
Content available remote Some remarks on Hausdorff gaps and automorphisms of P(ω)/fin
EN
We present, under the Continuum Hypothesis (CH), a construction of an automorphism of P(ω)/fin which maps a Hausdorff gap onto increasingly ordered gap of type (ω1, ω1) which is not a Hausdorff gap.
PL
Artykuł przedstawia, przy założeniu Hipotezy Continuum, konstrukcję automorfizmu algebry P(ω)/fin, który przeprowadza lukę Hausdorffa na lukę niemającą własności Hausdorffa.
EN
A new classification of arbitrary cellular automata (CA for short) in Z^d is studied considering the set (group) of all permutations of the neighborhood v and state set Q. Two CA (Z^d, Q, f_A, .A) and (Z^d, Q, f_B, v_B) are called automorphic, if there is a pair of permutationsπ&pfi" of v and Q, respectively, such that (f_B, vB) = ([formula] where v^π denotes a permutation of v and f*π denotes a permutation of arguments of local function f corresponding to v*π This automorphism naturally induces a classification of CA, such that it generally preserves the global properties of CA up to permutation. As a typical example of the theory, the local functions of 256 ECA (1- dimensional 3-nearest neighbors 2-states CA) are classified into 46 classes. We also give a computer test of surjectivity, injecitivity and reversibility of the classes.
3
Content available remote Identities with two automorphisms on semiprime rings
EN
In this paper we investigate identities with two automorphisms on semiprime rings. We prove the following result: Let T, S : R approaches R be automorphisms where R is a 2-torsion free semiprime ring satisfying the relation T(x)x = xS(x) for all x is an element of R. In this case the mapping x approaches T(x) - x maps R into its center and T = S.
4
Content available remote Invariance and Set-Theoretical Operations in First Order Structures
EN
We present a generalization of a theorem of Krasner showing how to construct relations invariant by automorphisms of a first order structure, by means of suitable set-theoretical operations.
5
Content available remote Identities with products of (alpha, beta)-derivations on prime rings
EN
The main purpose of this paper is to prove the following result. Let R be a noncommutative prime ring of characteristic different from two and let D and G = 0 be (\alpha, beta)-derivations of R into itself such that G commutes with alpha and beta. If [D{x), G(x)] = 0 holds for all x is an eleemnt of R then D = lambdaG where lambda is an element from the extended centroid of R.
6
Content available remote On alfa-derivations of prime and semiprime rings
EN
In this paper we investigate identities with alfa-derivations on prime and semiprime rings. We prove, for example, the following result. If D : R - R is an alfa-derivation of a 2 and 3-torsion free semiprime ring R such that [D(x},x2] = 0 holds, for all x is an element of R, then D maps R into its center. The results of this paper are motivated by the work of Thaheem and Samman [20].
7
Content available remote Free actions of semiprime rings with involution induced a derivation
EN
Let R be an associative ring. An element a is an element of R is said to be dependent of a mapping F : R -> R in case F (x) a = ax holds for all x is an element of R. A mapping F : R -> R is called a free action in case zero is the only dependent element of F. In this paper free actions of semiprime *- rings induced by a derivation are considered. We prove, for example, that in case we have a derivation D : R -> R, where R is a semiprime *-ring, then the mapping F defined by F(x) = D(x*) + D(x)*,x is an element of R, is a free action. It is also proved that any Jordan *-derivation on a 2-torsion free semiprime *-ring is a free action.
8
Content available remote On (α, β)-derivations of semiprime rings, II
9
Content available remote Centralizing mappings and derivations on semiprime rings
EN
In this paper we study some properties of centralizing mappings on semi-prime rings. The main purpose is to prove the result: Let -R be a semiprime ring and f an endomorphism of R, g an epimorphism of R such that the mapping x -> [f(x),g(x)] is central. Then [f(x),g(x)] = 0 holds for all x e R. We also establish some results about (alpha,beta)-derivations.
10
Content available remote On (alpha,beta)-derivations of semiprime rings
EN
We show that if alpha and beta are centralizing automorphisms and d a centralizing (alpha,beta)-derivation of a semiprime ring -R, then d is commuting. Some results on alpha-derivations and centralizing derivations of semiprime rings follow as applications of this result.
11
Content available remote Affine geometry of spine spaces
EN
The parallelity relation and the group of dilatations in the geometry of spine spaces are investigated. Fundamental theorems of affine geometry are proved and the analytical representation of dilatations is given.
12
Content available remote Nearaffine planes related to pseudo-ordered fields
EN
The constructions of affine planes and Minkowski planes related to pseudo-ordered fields are given in [3] and [2], respectively. We here give some analogous construction for nearaffine planes. Like before, we shall use some functions f, g and determine some conditions on f, g, necessary and suffcient to get the required plane. The Veblen postulate has a particular meaning in nearaffine planes, so it is also considered in the work. Some special cases like the field of the reals and finite fields of odd order are investigated, too. We give some examples of such nearaffine planes and consider their particular automorphisms. Every Minkowski plane related to pseudo-ordered field F determines a nearaffine plane connected with F [2, Proposition l, p. 187]. But only weaker version of the reciprocal statement is true, i.e. a nearaffine plane related to a pseudo-ordered field determines a hyperbola structure (i.e. Minkowski plane without touching axiom).
13
Content available remote Dynamical properties of automorphisms of minimal flows
EN
The dynamics of automorphisms of minimal flows are studied using a group which generalizes the Ellis group of the flow.
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