Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

Non Σn axiomatizable almost strongly minimal theories

1989 ◽  
Vol 54 (3) ◽  
pp. 921-927 ◽  
Author(s):  
David Marker
Keyword(s):  
Natural Number ◽  
Algebraic Closure ◽  
Predicate Symbol ◽  
Elementary Extension ◽  
Minimal Set ◽  
Categorical Theory ◽  
Minimal Theory ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Infinite Power

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

Minimal Sets on Graphs and Dendrites

2003 ◽  
Vol 13 (07) ◽  
pp. 1721-1725 ◽  
Author(s):  
Francisco Balibrea ◽  
Roman Hric ◽  
L'ubomír Snoha
Keyword(s):  
Topological Structure ◽  
Partial Characterization ◽  
Minimal Set ◽  
Full Characterization ◽  
Minimal Sets ◽  
Continuous Maps

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

One theorem of Zil′ber's on strongly minimal sets

1985 ◽  
Vol 50 (4) ◽  
pp. 1054-1061 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Main Part ◽  
Minimal Set ◽  
Minimal Sets ◽  
Image Position

AbstractSuppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,We prove the following theorems.Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

scholarly journals Quality of Minimal Sets of Prime Implicants of Boolean Functions

2017 ◽  
Vol 63 (2) ◽  
pp. 165-169
Author(s):  
V. C. Prasad
Keyword(s):  
Power Consumption ◽  
Boolean Function ◽  
Boolean Functions ◽  
New Technique ◽  
Minimal Set ◽  
Reduce Power Consumption ◽  
Minimal Sets ◽  
Prime Implicants

Abstract Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

The classification of small weakly minimal sets. III: Modules

1988 ◽  
Vol 53 (3) ◽  
pp. 975-979 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Finite Field ◽  
Algebraic Closure ◽  
Morley Rank ◽  
Minimal Sets

AbstractTheorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let A = acl(∅) ⋂ M and I = {r ∈ R: rM ⊂ A}. Notice that I is an ideal.(i) F = R/Iis a finite field.(ii) Suppose that a, b0,…,bn, ∈ M and . Then there are s, ri ∈ R, i ≤ n, such that sa + Σi≤nribi ∈ A and s ∉ I.It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtainTheorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M).

Regular minimal sets over the circle and the ellis minimal set

1972 ◽  
Vol 6 (1-2) ◽  
pp. 145-163
Author(s):  
Richard Freiman
Keyword(s):  
Minimal Set ◽  
Minimal Sets

scholarly journals DMP in strongly minimal sets

2007 ◽  
Vol 72 (3) ◽  
pp. 1019-1030 ◽  
Author(s):  
Assaf Hasson ◽  
Ehud Hrushovski
Keyword(s):  
Finite Cover ◽  
Minimal Set ◽  
Minimal Sets ◽  
Minimal Theory

AbstractWe construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T, generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

scholarly journals On Finite Groups With ‘Hidden’ Primes

1971 ◽  
Vol 12 (3) ◽  
pp. 287-300 ◽  
Author(s):  
L. G. Kovács ◽  
Joachim Neubüser ◽  
B. H. Neumann
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Finite Groups ◽  
Infinite Order ◽  
Proper Subset ◽  
Minimal Set ◽  
Minimal Sets ◽  
Starting Point

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G, must there be a minimal set of generators of G that contains an element whose order is divisible by p? A set of generators of G is minimal if no set with fewer elements generates G. A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

scholarly journals Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

2021 ◽  
pp. 117-137
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Systematic Investigation ◽  
Exact Matching ◽  
Minimal Set ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Switching Functions ◽  
Programming Techniques ◽  
Karnaugh Map

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

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