Discriminator Varieties Research Articles

Free algebras in discriminator varieties

We investigate V-free algebras onn generators,F n =F r (V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindricalgebras. Sample questions are: (a) IsFn+1 embeddable inF n ? (b) DoesF n contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.

Rich countable epimorphism skeletons of discriminator varieties

Rich countable epimorphism skeletons of discriminator varieties

Invariant subspaces of some functional spaces on the group of motions of the Euclidean plane

i. A. G. Pinus, "On the relations of epimorphism and embeddability on congruence-distributive varieties," Algebra Logika, 24, NO. 5, 588-607 (1985). 2. A. G. Pinus, Congruence-Modular Varieties of Algebras [in Russianl, Irkutsk State Univ. (1986). 3. A. G. Pinus, "On simple epimorphism skeletons of congruence-distributive varieties," Izv. Vyssh. Uchebn. Zaved., Mat., No. ii, 67-70 (1987). 4. A. G. Pinus, "On coverings in epimorphism skeletons of varieties of algebras," Algebra Logika, 27, No. 3, 316-326 (1988). 5. A. G. Pinus, "ConGruence-distributive varieties of algebras," Progress in Science. Algebra, Geometry, and Toplogy [in Russian], Vol. 26, Bsesoyuz. Inst. Nauch. i Tekh. Informatsii, Moscow (1988), pp. 45-83. 6. A. G. Pinus, "On the number of incomparables in countable epimorphism skeletons of discriminator varieties," Algebra Logika, 2_88, No. 3, 311-323 (1989). 7. G. Gritzer, Universal Algebra, 2nd Ed., Springer-Verlag, Berlin-Heidelberg (1979). 8. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York-Heidelberg-Berlin (1981). H. Werner, Discriminator Algebras, Akademie Verlag, Berlin (1978). G. A. Fraser and A. Horn, "Congruence relations in direct product," Proc. Am. Math. Soc., 26, No. 3, 390-394 (1970). .

Countable embeddability skeletons of discriminator varieties

Countable embeddability skeletons of discriminator varieties

On the number of discriminator varieties incomparable in countable epimorphism skeletons

On the number of discriminator varieties incomparable in countable epimorphism skeletons

The Fraser-Horn and Apple properties

We consider varieties V \mathcal {V} in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with V \mathcal {V} a family of derived varieties, d ( V ) d(\mathcal {V}) : a variety in d ( V ) d(\mathcal {V}) is generated by algebras A {\mathbf {A}} where the universe of A {\mathbf {A}} consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra B ∈ V {\mathbf {B}} \in \mathcal {V} and the operations of A {\mathbf {A}} are those of B {\mathbf {B}} that preserve this congruence class. We also consider the prime variety of V \mathcal {V} , denoted V 0 {\mathcal {V}_0} , generated by all finite simple algebras in V \mathcal {V} . We show how the structure of finite algebras in V \mathcal {V} is determined to a considerable extent by V 0 {\mathcal {V}_0} and d ( V ) d(\mathcal {V}) . In particular, the free V \mathcal {V} -algebra on n n generators, F V ( n ) {{\mathbf {F}}_\mathcal {V}}(n) , has as many directly indecomposable factors as F V 0 ( n ) {{\mathbf {F}}_{{\mathcal {V}_0}}}(n) and the structure of these factors is determined by the varieties d ( V ) d(\mathcal {V}) . This allows us to produce in many cases explicit formulas for the cardinality of F V ( n ) {{\mathbf {F}}_\mathcal {V}}(n) . Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.

Quantifier elimination in discriminator varieties

Caracterisation des classes d'elements d'une variete de discriminateurs admettant l'elimination des quantificateurs en termes de classes d'elements simples

Finitely generic models of TUH, for certain model companionable theories T

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Iterated discriminator varieties have undecidable theories

Iterated discriminator varieties have undecidable theories

Equationally Complete Discriminator Varieties of Groupoids

Equationally Complete Discriminator Varieties of Groupoids

Equationally complete discriminator varieties of groupoids

J. Kalicki proved that there are continuum many equationally complete varieties of groupoids. In this note we give a constructive proof of this by defining a countable partial groupoid which has continuum many completions such that each completion generates an equationally complete variety, and no two distinct completions generate the same variety. Moreover, the variety generated by all the completions is a discriminator variety, and every nontrivial groupoid in this variety is cancellative but not a quasigroup; this answers a question of R. Padmanabhan. A. D. Bol’bot proved a similar result for loops, but his computations are more difficult since his varieties are not discriminator varieties.

Expanding varieties by monoids of endomorphisms

The purpose of this paper is to start a general investigation of the varieties "P'(M) obtained by expanding a variety T" by a monoid of endomorphisms M. This construction was used in [3] to manufacture the first example of a variety with a decidable theory and not of the form (discriminator)| It also plays a key role in Baur's papers [1], [2] on the first-order theory of Abelian groups with distinguished subgroups. In the first section a few basic results are presented. In the second section we describe exactly when 7/'(M) is a discriminator variety, generalizing the treatment of Nsg(G) given in [3]. The final section is devoted to Abelian varieties and the corresponding varieties of modules.

Finitely Boolean Representable Varieties

This paper gives a short, elementary proof of a result of Burris and McKenzie [2] stating that each variety Boolean representable by a finite set of finite algebras is the join of an abelian and a discriminator variety. An example showing that the Boolean product operator ${\Gamma ^a}$ is not idempotent is included as well.

Finitely Boolean representable varieties

This paper gives a short, elementary proof of a result of Burris and McKenzie [2] stating that each variety Boolean representable by a finite set of finite algebras is the join of an abelian and a discriminator variety. An example showing that the Boolean product operator Γ a {\Gamma ^a} is not idempotent is included as well.

Decidable varieties with modular congruence lattices

For a large collection of varieties we show that if the first-order theory of such a variety is decidable then the variety decomposes into the product of two well-known highly specialized varieties. For many varieties the decidability question then reduces to a decidability question about modules. A variety is a class of (abstract) algebras (belonging to some language) closed under the formation of direct products, subalgebras and homomorphic images. A variety 1/ is locally finite if every finitely-generated member of (/ is finite. A class of algebras K generates a variety 1/ if 1/ is the smallest variety containing K—then we say that \J is generated by K, written 1/ = V{K). A variety is finitely generated if it is generated by finitely many finite algebras, or equivalently by a single finite algebra. The kernel of a homomorphism is called a congruence, and the congruences of any algebra form a lattice. A variety is congruence modular, or we prefer to say just modular, if the lattice of congruences of every algebra in the variety satisfies the modular law. (Most of the well-studied varieties are modular; for example varieties of groups, rings, modules and lattices are modular. However,the variety of semigroups is not modular.) A variety \] is a product of two subvarieties | / j , |/2 ifl / j U l/2 generates (/ and there is a term b(x, y) such that M x \= b{x, y) = x, l/2 |= b(x, y) = y. If this is so we write 1/ = 1/ x <£) 1/ 2 , and then for every algebra A in 1/ there are (up to isomorphism) unique algebras Ax G (/p A2 G (/2 such that A = Ax x A2. A class K of first-order structures has a decidable theory if there is an effective procedure to determine precisely which first-order sentences are true of every member of K. A variety 1/ is a discriminator variety if it is generated by a set K for which there exists a ternary term t(xf y, z) such that K satisfies t(x, yy z) = x if x =£ y = z if x — y. In everyday mathematics such varieties appear only as highly specialized varieties of rings, or varieties associated with algebraic logics. Received by the editors December 29, 1980. 1980 Mathematics Subject Classification. Primary 03B25, 08B10, 08A05.