The description of categorical quasivarieties

Palyutin, E. A.

doi:10.1007/bf01668422

Cited by 31 publications

(7 citation statements)

References 4 publications

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“…The description of a variety by means of its fine spectrum relates to a wellknown description by Palyutin [95] and Givant [55] of categorical varieties. On the other hand, Baldwin and McKenzie [7] have described all possible types of function from the fine spectrum (onto infinite cardinal numbers) for congruence-modular varieties, and Palyutin [97] has done the same for arbitrary varieties.…”

Section: Categories and Spectra Of Varietiesmentioning

confidence: 97%

Boolean constructions in universal algebra

Pinus¹

1992

Russ. Math. Surv.

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Section: Categories and Spectra Of Varietiesmentioning

confidence: 97%

Boolean constructions in universal algebra

Pinus¹

1992

Russ. Math. Surv.

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“…A class of algebras is ω-categorical if up to isomorphism there is a single countably infinite algebra in the class. What is perhaps more interesting is that our results can be used to provide another proof of the classification of ω-categorical varieties [4,5,8,14,15], since it is not difficult to show that such a variety must be locally finite, abelian and minimal (see Theorem 4.1 of [8]). …”

Section: Corollary 55 a Minimal Locally Finite Affine Variety Is Cmentioning

confidence: 98%

Minimal sets and varieties

Kearnes

Kiss

Valeriote

1998

Trans. Amer. Math. Soc.

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Abstract. The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

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“…In [17] Paljutin shows that "There exists a finitely axiomatizable, not locally finite categorical quasi-variety if and only if one of the following conditions hold: 1) there exists an infinite finitely presented ring which is a division ring;…”

Section: Application To Finite Axiomatizabilitymentioning

confidence: 99%

Finitely axiomatizable strongly minimal groups

Blossier¹,

Bouscaren²

2012

J. symb. log.

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We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasi-endomorphisms of G must be an infinite finitely presented ring.Questions about finite axiomatizability of first order theories are nearly as old as model theory itself and seem at first glance to have a fairly syntactical flavor. But it was in order to show that totally categorical theories cannot be finitely axiomatized that, in the early eighties, Boris Zilber started developing what is now known as "Geometric stability theory". Indeed, as is often the case, in order to answer such a question, one needs to develop a fine analysis of the structure of models in the class involved and to understand exactly how each model is constructed.The easiest way to force a structure to be infinite by one first order sentence is to impose an ordering without end points, or a dense ordering, thus making the structure unstable. It was hence rather natural to wonder about theories at the other extremity of the stability spectrum, and in the early 60's the question was posed whether there existed finitely axiomatizable totally categorical theories or simply uncountably categorical theories [21], [16].Each model of a totally categorical theory is prime over a strongly minimal set. It is not too difficult to see that a totally categorical strongly minimal set cannot be finitely axiomatizable ([14]). Much more complicated, the proof of the non finite axiomatizability for the whole class goes through a characterization of the geometries associated to totally categorical strongly minimal sets (locally modular and locally finite) and then an analysis of how any model is "built" around the strongly minimal set ([22], [23] and [6] where the result is proved for all ω-stable ω-categorical theories).Around the same time as Zilber's negative answer for the totally categorical case, Peretjat'kin produced an example of a finitely axiomatized ℵ 1 -categorical theory [18]. This example was in the following years simplified by Baisalov (see [9, 12.2, Example 5]). This final example has Morley Rank equal to 2, thus still leaving open the question of the existence of a finitely axiomatizable strongly minimal set (Morley rank and degree equal to 1). Furthermore all the known examples of finitely axiomatizable ℵ 1 -categorical theories are rather similar and constructed around a strongly minimal set with trivial 1

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The description of categorical quasivarieties

Cited by 31 publications

References 4 publications

Boolean constructions in universal algebra

Boolean constructions in universal algebra

Minimal sets and varieties

Finitely axiomatizable strongly minimal groups

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