Existentially closed algebras and boolean products

1988 ◽  
Vol 53 (2) ◽  
pp. 571-596
Author(s):  
Herbert H. J. Riedel
Keyword(s):  
Sufficient Condition ◽  
Subdirectly Irreducible ◽  
Universal Class ◽  
Model Companion ◽  
Universal Horn Class ◽  
Existentially Closed ◽  
Boolean Product ◽  
Boolean Products ◽  
Simple Algebras ◽  
Subdirectly Irreducible Algebras

AbstractA Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

Regular double p-algebras

2019 ◽  
Vol 69 (1) ◽  
pp. 15-34 ◽  
Author(s):  
M. E. Adams ◽  
Hanamantagouda P. Sankappanavar ◽  
Júlia Vaz de Carvalho
Keyword(s):  
Free Algebra ◽  
Explicit Description ◽  
Priestley Duality ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
Equational Basis ◽  
Proper Subvariety ◽  
Simple Algebras ◽  
Countably Infinite ◽  
Subdirectly Irreducible Algebras

Abstract In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

Model companions for finitely generated universal Horn classes

1984 ◽  
Vol 49 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Stanley Burris
Keyword(s):  
Finitely Generated ◽  
Model Companion ◽  
Universal Horn Class ◽  
Finite Structure ◽  
Finite Structures ◽  
Structure Theorems ◽  
Joint Embedding ◽  
Joint Embedding Property ◽  
Primitive Recursive

AbstractIn an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ0-categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal Horn class is generated by a single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups.

scholarly journals The subdirect decomposition theorem for classes of structures closed under direct limits

1980 ◽  
Vol 30 (2) ◽  
pp. 171-179 ◽  
Author(s):  
Xavier Caicedo
Keyword(s):  
Subdirect Product ◽  
Decomposition Theorem ◽  
Subdirectly Irreducible ◽  
Subdirect Decomposition ◽  
Direct Limits ◽  
Subdirectly Irreducible Algebras

AbstractBy a theorem of G. Birkhoff, every algebra in an equationally defined class of algebras K is a subdirect product of subdirectly irreducible algebras of K. In this paper we show that this result is true for any class of structures. not necessarily algebraic, closed under isomorphisms and direct limits. Quasivarieties in the sense of Malcev are examples of such classes of structures. This includes Birkhoffs result as a particular case.

scholarly journals Injectives in some small varieties of ockham algebras

1984 ◽  
Vol 25 (2) ◽  
pp. 183-191 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Duality Theory ◽  
Free Algebras ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Double Negation ◽  
Topological Duality ◽  
De Morgan Algebras ◽  
Ockham Algebras ◽  
The One ◽  
Subdirectly Irreducible Algebras

The study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. On the one hand, this class of algebras simultaneously abstracts de Morgan algebras and Stone algebras while, on the other hand, it has relevance to propositional logics lacking both the paradoxes of material implication and the law of double negation. Subsequently, these algebras were baptized distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [13]. In an elegant paper [9], M. S. Goldberg extended this theory and, amongst other things, described the free algebras and the injective algebras in those subvarieties of the variety 0 of distributive Ockham algebras which are generated by a single finite subdirectly irreducible algebra. Recently, T. S. Blyth and J. C. Varlet [4] explicitly described the subdirectly irreducible algebras in a small subvariety MS of 0 while in [2] the order-topological results of Goldberg were applied to accomplish the same objective for a subvariety k1.1 of 0 which properly contains MS. The aim, here, is to describe explicitly the injective algebras in each of the subvarieties of the variety MS. The first step is to draw the Hasse diagram of the lattice AMS of subvarieties of MS. Next, the results of Goldberg are applied to describe the injectives in each of the join irreducible members of AMS. Finally, this information, in conjunction with universal algebraic results due to B. Davey and H. Werner [8], is applied to give an explicit description of the injectives in each of the join reducible members of AMS.

Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups

1996 ◽  
Vol 61 (1) ◽  
pp. 277-284 ◽  
Author(s):  
Anatole Khelif
Keyword(s):  
Open Problem ◽  
General Framework ◽  
Direct Proof ◽  
Model Companion ◽  
Standard Methods ◽  
Division Rings ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Constructible Sets

AbstractWe prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.Our main result uses constructible sets and can be put in an abstract general framework.Comparison with the standard methods which use forcing (cf. [4]) is sketched.

scholarly journals Existentially closed models of the theory of artinian local rings

1999 ◽  
Vol 64 (2) ◽  
pp. 825-845 ◽  
Author(s):  
Hans Schoutens
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Model Companion ◽  
Closed Model ◽  
Gorenstein Rings ◽  
Existentially Closed ◽  
Finite Set ◽  
Residue Fields

AbstractThe class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

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