Yazar "Belegradek, O" seçeneğine göre listele
Listeleniyor 1 - 5 / 5
Sayfa Başına Sonuç
Sıralama seçenekleri
Öğe Coset-minimal groups(Elsevier Science Bv, 2003) Belegradek, O; Verbovskiy, V; Wagner, FOA totally ordered group G (possibly with extra structure) is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G boolean OR {+/-infinity}. Continuing work in Belegradek et al. (J. Symbolic Logic 65(3) (2000) 1115) and Point and Wagner (Ann. Pure Appl. Logic 105(1-3) (2000) 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian, an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any pure ordered abelian group of finite regular rank is ultimately coset-minimal and has the exchange property; moreover, every definable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classified. In a discrete coset-minimal group every definable unary function is piece-wise linear (this improves a result in Point and Wagner (Ann. Pure Appl. Logic 105(1-3) (2000) 261), where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property (which is false in the discrete case (M.S.R.I., preprint series, 1998-051)); moreover, any definable unary function is piecewise linear, except possibly for finitely many cosets of the smallest definable convex nonzero subgroup. Finally, we give some examples and open questions. (C) 2002 Elsevier Science B.V. All rights reserved.Öğe Discriminating and square-like groups(Walter De Gruyter Gmbh, 2004) Belegradek, OA group is discriminating if and only if it discriminates its direct square, and square-like if and only if it is universally equivalent to its direct square. It is known that any discriminating group is square-like. These notions were introduced and studied in a series of papers by Baumslag, Myasnikov and Remeslennikov and by Fine, Gaglione, Myasnikov and Spellman. We prove that any square-like group is elementarily equivalent to a countable discriminating group. This answers a question of the second group of authors. We provide an explicit universal - existential axiom system for the class of square-like groups. We show that the theory of the class of discriminating groups is computably enumerable but undecidable. We give a criterion for determining whether a group is discriminating. We propose a construction method for discriminating groups and use it to construct in various group varieties many discriminating non-abelian groups that do not embed their squares. We construct square-like, nondiscriminating nilpotent p-groups of arbitrary nilpotency class; all previously known square-like, non-discriminating groups were abelian.Öğe Finitely determined members of varieties of groups and rings(Academic Press Inc, 2000) Belegradek, OA finitely generated algebra A in a variety V is called finitely determined in V if there exists a finite V-consistent set of equalities and inequalities in an alphabet containing the generating set of A, which, together with the identities of V, yields all relations and non-relations of A. Obviously, if the equational theory of V is recursively enumerable then any finitely determined algebra in V has solvable word problem. The known algebraic characterizations of groups and semigroups with solvable word problem imply that in the varieties of all groups and all semigroups the members with solvable word problem are finitely determined. We construct a finitely generated center-by-metabelian group with solvable word problem, which is not finitely determined in every group variety V with ZU(2) subset of or equal to V subset of or equal to U-3. We show that every extension of a finitely generated abelian group by a finite group from a variety W is finitely determined in every variety V superset of or equal to ZU(2)W. However, in any abelian-by-nilpotent variety no infinite group is finitely determined; moreover, in every variety, in which all finitely presented algebras are residually finite, each finitely determined algebra is finite. In the variety of all associative linear algebras over a finitely generated field every member with solvable word problem is finitely determined. We construct an example, which shows that for the variety of all associative rings it is not true; however, in this variety each torsion-free member with solvable word problem is finitely determined. (C) 2000 Academic Press.Öğe Quasi-o-minimal structures(Assn Symbolic Logic Inc, 2000) Belegradek, O; Peterzil, Y; Wagner, FA structure (M, <....) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal: one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use ii to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1.Öğe Semi-bounded relations in ordered modules(Cambridge Univ Press, 2004) Belegradek, OA relation on a linearly ordered structure is called semi-bounded if it is definable in an expansion of the structure by bounded relations. We study ultimate behavior of semi-bounded relations in an ordered module M over an ordered commutative ring R such that M/rM is finite for all nonzero r is an element of R. We consider M as a structure in the language of ordered R-modules augmented by relation symbols for the submodules rM, and prove several quantifier elimination results for semi-bounded relations and functions in M. We show that these quantifier elimination results essentially characterize the ordered modules M with finite indices of the submodules rM. It is proven that (1) any semi-bounded k-ary relation on M is equal, outside a finite union of k-strips, to a k-ary relation quantifier-free definable in M, (2) any semi-bounded function from M-k to M is equal, outside a finite union of k-strips, to a piecewise linear function, and (3) any semi-bounded in M endomorphism of the additive group of M is of the form x --> sigmax, for some sigma from the field of fractions of R.
