DETERMINING WHETHER ${\mathsf V}({\bf A})$ HAS A MODEL COMPANION IS UNDECIDABLE

2004 ◽  
Vol 14 (03) ◽  
pp. 325-355
Author(s):  
ROSS WILLARD
Keyword(s):  
Finite Algebra ◽  
Equational Class ◽  
Model Companion ◽  
Finite Language

Using techniques pioneered by R. McKenzie, we prove that there is no algorithm which, given a finite algebra in a finite language, determines whether the variety (equational class) generated by the algebra has a model companion. In particular, there exists a finite algebra such that the variety it generates has no model companion; this answers a question of Burris and Werner from 1979.

scholarly journals A finite basis theorem for residually finite, congruence meet-semidistributive varieties

2000 ◽  
Vol 65 (1) ◽  
pp. 187-200 ◽  
Author(s):  
Ross Willard
Keyword(s):  
Finite Algebra ◽  
Finite Basis ◽  
Finitely Based ◽  
Mal’Cev Condition ◽  
Residually Finite ◽  
Basis Theorem ◽  
Finite Language

AbstractWe derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m < ω and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m.

scholarly journals Polynomial-time tests for difference terms in idempotent varieties

2019 ◽  
Vol 29 (06) ◽  
pp. 927-949
Author(s):  
William DeMeo ◽  
Ralph Freese ◽  
Matthew Valeriote
Keyword(s):  
Polynomial Time ◽  
Finite Algebra ◽  
Finite Language ◽  
Difference Term ◽  
Practical Question

We consider the following practical question: given a finite algebra [Formula: see text] in a finite language, can we efficiently decide whether the variety generated by [Formula: see text] has a difference term? We answer this question (positively) in the idempotent case and then describe algorithms for constructing difference term operations.

The structure of algebraically and existentially closed Stone and double Stone algebras

1989 ◽  
Vol 54 (2) ◽  
pp. 363-375 ◽  
Author(s):  
David M. Clark
Keyword(s):  
Finite Algebra ◽  
Amalgamation Property ◽  
Explicit Construction ◽  
Structural Description ◽  
Model Companion ◽  
Existentially Closed ◽  
Boolean Spaces ◽  
Countably Infinite ◽  
Model Completion ◽  
Primitive Recursive

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

scholarly journals Free algebra structure: categorical algebras

1970 ◽  
Vol 3 (2) ◽  
pp. 207-215 ◽  
Author(s):  
H. G. Moore
Keyword(s):  
Positive Integer ◽  
Free Algebra ◽  
Finite Algebra ◽  
Commutative Rings ◽  
Equational Class ◽  
Algebra Structure ◽  
Direct Power ◽  
Universal Algebras ◽  
Boolean Rings ◽  
Categorical Algebra

One of the more important concepts in the study of universal algebras is that of a free algebra. It is our purpose in this communication to describe the structure of the free algebra Kk() of k generators (k a positive integer) determined by a categorical algebra, and to indicate how this information encompasses results in such diverse areas as the study of Post algebras, boolean rings, p-rings, pk-rings, finite commutative rings with unity, etc.A finite algebra is called categorical if every algebra in its equational class is isomorphic to a sub-direct power of A. If has n elements, permutable identities, no non-identical automorphism and exactly m distinct one-element subalgebras, then .

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

scholarly journals Stone algebras form an equational class: (Remarks on Lattice Theory III)

1969 ◽  
Vol 9 (3-4) ◽  
pp. 308-309 ◽  
Author(s):  
G. Grätzer
Keyword(s):  
Lattice Theory ◽  
Equational Class ◽  
Distributive Lattices ◽  
Image Position

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

scholarly journals A semi-finite algebra associated to a subfactor planar algebra

2011 ◽  
Vol 261 (5) ◽  
pp. 1345-1360 ◽  
Author(s):  
A. Guionnet ◽  
V. Jones ◽  
D. Shlyakhtenko
Keyword(s):  
Finite Algebra ◽  
Planar Algebra
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