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In this article we define a new form of unipotence in groups of finite Morley rank which extends Burdges unipotence to any characteristic. In particular, we show that every connected solvable group of finite Morley rank G has a definable connected subgroup H whose derived subgroup H' is a good unipotent subgroup of finite Morley rank. [ABSTRACT FROM AUTHOR]

This paper is based on the concept of roughness in JU-algebras and related terminologies. In this paper, we study lower and upper approximations of JU-subalgebras and JU-ideals. Moreover, we study rough sets and JU-algebras with their weak and strong ideals and prove that the lower/upper approximation of JU-subalgebra (resp., JU-ideals) is a JU-subalgebra (resp., JU-ideals). Furthermore, we show some related results. [ABSTRACT FROM AUTHOR]

We generalize Kada's definable strengthening of Dilworth's characterization of the class of quasi-orders admitting an antichain of a given finite cardinality. [ABSTRACT FROM AUTHOR]

We study t-structures with Grothendieck hearts on compactly generated triangulated categories T that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of T in terms of directed homotopy colimits. For a left nondegenerate t-structure t = (U , V) on T , we show that V is definable if and only if t is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to t being homotopically smashing and to t being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of t are determined by purity conditions on the associated partial cosilting object. [ABSTRACT FROM AUTHOR]

We prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E 2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms. [ABSTRACT FROM AUTHOR]

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to the 2-category of fp-hom-closed definable additive categories satisfying an exactness criterion. For a fixed finitely accessible category C with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of C , the Serre tensor-ideals of C fp - mod and the closed subsets of a Ziegler-type topology. For a skeletally small preadditive category A with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of Mod - A and the definable tensor-ideals of A - Mod. [ABSTRACT FROM AUTHOR]

Let (U';C') be a subspace of a covering approximation space (U;C) and X⊂U'. In this paper, we show that and B'(X)⊂B(X)∩U'. Also, iff (U;C) has Property Multiplication. Furthermore, some connections between outer (resp. inner) definable subsets in (U;C) and outer (resp. inner) definable subsets in (U';C') are established. These results answer a question on covering approximation subspace posed by J. Li, and are helpful to obtain further applications of Pawlak rough set theory in pattern recognition and artificial intelligence.

Let M be a fine structural mouse and let F ∈ M be such that M ⊨" F is a total extender" and (M | | lh (F) , F) is a premouse. We show that it follows that F ∈ E M , where E M is the extender sequence of M. We also prove generalizations of this fact. Let M be a premouse with no largest cardinal and let Σ be a sufficient iteration strategy for M. We prove that if M knows enough of Σ ↾ M then E M is definable over the universe ⌊ M ⌋ of M , so if also ⌊ M ⌋ ⊨ ZFC then ⌊ M ⌋ ⊨ " V = HOD ". We show that this result applies in particular to M = M nt | λ , where M nt is the least non-tame mouse and λ is any limit cardinal of M nt. We also show that there is no iterable bicephalus (N , E , F) for which E is type 2 and F is type 1 or 3. As a corollary, we deduce a uniqueness property for maximal L [ E ] constructions computed in iterable background universes. [ABSTRACT FROM AUTHOR]

Tree paths are investigated using first-order logic. The following results are obtained: (i) every definable path can be defined by a first-order formula using at most one parameter chosen from the path itself; (ii) a canonical representation of the formulas that define definable paths is obtained; and (iii) every tree that has only finitely many paths that are not definable is n-equivalent to a tree of which all paths are definable. Moreover, a certain property that might be expected to hold, involving the transfer of n-equivalence between trees, is shown not to be true. [ABSTRACT FROM AUTHOR]

Let G be an arbitrary group. We show that if the Fitting subgroup of G is nilpotent then it is definable. We prove also that the class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given; that is for the subgroup generated by all normal soluble subgroups of G. [ABSTRACT FROM AUTHOR]

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Vélez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Vélez. [ABSTRACT FROM AUTHOR]

Abstract: We discover geometric properties of certain definable sets over non-Archimedean valued fields with analytic structures. Results include a parameterized smooth stratification theorem and the existence of a bound on the piece number of fibers for these sets. In addition, we develop a dimension theory for these sets and also for the formulas which define them. [Copyright &y& Elsevier]

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts. [ABSTRACT FROM AUTHOR]