Finite Varieties Research Articles

On Finite Basis Property for Joins of Varieties of Associative Rings

If all finitely generated rings in a variety of associative rings satisfy the ascending chain condition on two-sided ideals, the variety is called locally weak noetherian. If there is an upper bound on nilpotency indices of nilpotent rings in a variety, the variety is called a finite index variety. We prove that the join of a finitely based locally weak noetherian variety and a variety of finite index is also finitely based and locally weak noetherian. One consequence of this result is that if an associative ring variety is connected by a finite path in the lattice of all associative ring varieties to a finitely based locally weak noetherian variety then such variety is also finitely based and locally weak noetherian.

Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.

On identities of finite involution semigroups

We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.

Varieties of rings, where all subdirectly irreducible finite rings are Armendariz ones

In this paper we study varieties of rings, where all subdirectly irreducible finite rings are Armendariz. We also describe the locally finite varieties of Armendariz rings.

Existence theorems for weakly symmetric operations

A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this

Definable principal congruences and solvability

We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

The structure of locally finite varieties with polynomially many models

We prove that a locally finite variety has at most polynomially many (in k k ) non-isomorphic k k –generated algebras if and only if it decomposes into a varietal product of an affine variety over a ring of finite representation type, and a sequence of strongly Abelian varieties equivalent to matrix powers of varieties of H H -sets, with constants, for various finite groups H H .

Gödel algebras free over finite distributive lattices

Gödel algebras form the locally finite variety of Heyting algebras satisfying the prelinearity axiom ( x → y ) ∨ ( y → x ) = ⊤ . In 1969, Horn proved that a Heyting algebra is a Gödel algebra if and only if its set of prime filters partially ordered by reverse inclusion– i.e. its prime spectrum–is a forest. Our main result characterizes Gödel algebras that are free over some finite distributive lattice by an intrisic property of their spectral forest.

Semidefinite representations for finite varieties

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x 1, . . . ,x n ]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality.

Locally finite varieties of Heyting algebras

We show that for a variety $$ \mathcal{V} $$ of Heyting algebras the following conditions are equivalent: (1) $$ \mathcal{V} $$ is locally finite; (2) the $$ \mathcal{V} $$ -coproduct of any two finite $$ \mathcal{V} $$ -algebras is finite; (3) either $$ \mathcal{V} $$ coincides with the variety of Boolean algebras or finite $$ \mathcal{V} $$ -copowers of the three element chain $$ {\text{3}} \in \mathcal{V} $$ are finite. We also show that a variety $$ \mathcal{V} $$ of Heyting algebras is generated by its finite members if, and only if, $$ \mathcal{V} $$ is generated by a locally finite $$ \mathcal{V} $$ -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: $$ \mathcal{V} $$ is finitely generated if, and only if, $$ \mathcal{V} $$ is residually finite.

The triangular principle is equivalent to the triangular scheme

The triangular scheme is a diagrammatic characterization of congruence join-semidistributivity. The triangular principle is a variant of this condition, where one of the congruences is replaced by a tolerance. This paper contains two proofs showing that the triangular principle and the triangular scheme are equivalent for varieties. The first one is a routine argument using tame congruence theory, and works only for locally finite varieties. The second proof is an elementary, but nontrivial calculation with terms, which works for arbitrary varieties. This yields a stronger Mal’tsev-condition characterizing congruence join-semidistributivity than the one obtained from the triangular scheme.

A new lattice construction

We introduce a new construction for orders and lattices. This construction is used to create large locally finite lattice varieties with uncountably many subvarieties.

On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis–Meakin expansion M ( H ) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin–Zelmanov varieties K n of all locally finite groups of exponent dividing n.

Finite posets and topological spaces in locally finite varieties

We prove that if a finite connected poset admits an order-preserving Taylor operation, then all of its homotopy groups are trivial. We use this to give new characterisations of locally finite varieties omitting type 1 in terms of the posets (or equivalently, finite topological spaces) in the variety. Similar variants of other omitting-type theorems are presented. We give several examples of posets that admit various types of Taylor operations; in particular, we exhibit a topological space which is not an H-space but is compatible with a set of non-trivial identities, answering a question of W. Taylor.

The Berman conjecture is true for nilpotent extensions of regular semigroups

Let S be any semigroup. A term operation of S is an n-ary operation of S which is induced by some nonempty word w by substitution of letters. Such an operation is essentially n-ary if it depends on all of its variables. For n ≥ 1, we denote the set of all essentially n-ary term operations of S by ES n , while ES 0 is the set of all constant unary term operations of S. Let pn(S) = |ES n | for all n ≥ 0. It is easy to see that the sequence pn(S), n ≥ 0, consists entirely of finite cardinals if and only if S generates a locally finite semigroup variety. Of course, all these concepts can be generalized for arbitrary algebras, cf. the survey paper by Gratzer and Kisielewicz [7]. The theory of pn-sequence of general algebras was founded by E. Marczewski and his ‘Wroc law School’ back in the sixties (see [9]). They identified as the main goal of this theory to characterize all sequences of non-negative integers which are representable as the pn-sequence of some algebra (or some particular kind of algebra). Later on, most of the research was limited to finite algebras. One of the most intriguing hypotheses in this direction was given by J. Berman, who conjectured in [1] that the pn-sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. While this assertion, referred to as the Berman conjecture, is easily shown to be true for finite monoids, groups, rings, modules, lattices, Boolean algebras, etc., R. Willard refuted this conjecture in [10] by constructing a very nice counterexample, a 4-element algebra with finitely many basic operations whose pn-sequence is (0, 3, 2, 5, 4, 7, 6, 9, 8, . . .). On the other hand, there are several papers (co-authored by the first named author of this note) investigating the (still open) question whether the Berman conjecture holds in the class of all finite semigroups. Namely, in [5] all finite semigroups whose pn-sequences have constant bounds are determined. In [6],

Interpretability Types for Regular Varieties of Algebras

It is proved that for every regular variety V of algebras, an interpretability type [V] in the lattice \({\mathbb{L}^{\operatorname{int} }} \) is primary w.r.t. intersection, and so has at most one covering. Moreover, the sole covering, if any, for [V] is necessarily infinite. For a locally finite regular variety V, [V] has no covering. Cyclic varieties of algebras turn out to be particularly interesting among the regular. Each of these is a variety of n-groupoids (A; f) defined by an identity \(f(x_1 ,...,x_n ) = f(x_{\lambda (1)} ,...,x_{\lambda (n)} )\), where λ is an n-cycle of degree n ≥ 2. Interpretability types of the cyclic varieties form, in \({\mathbb{L}^{\operatorname{int} }} \), a subsemilattice isomorphic to a semilattice of square-free natural numbers n ≥ 2, under taking m ∨ n=[m,n] (l.c.m.).

Unification, finite duality and projectivity in varieties of Heyting algebras

We investigate finitarity of unification types in locally finite varieties of Heyting algebras, giving both positive and negative results. We make essential use of finite dualities within a conceptualization for E-unification theory (J. Logic and Computation 7(6) (1997) 733–752) relying on the algebraic notion of a projective object.

Finite index subgroups in profinite groups

We prove that every subgroup of finite index in a ( topologically) finitely generated profinite group is open. This implies that the topology in such a group is uniquely determined by the group structure. The result follows from a ‘uniformity theorem’ about finite groups: given a group word w that defines a locally finite variety and a natural number d, there exists f= f w ( d) such that in every finite d-generator group G, each element of the verbal subgroup w( G) is a product of f w-values. Similar methods show that in a finite d-generator group, each element of the derived group is a product of g( d) commutators; this implies that the (abstract) derived group in any finitely generated profinite group is closed. To cite this article: N. Nikolov, D. Segal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Q -universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.¶ It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal.

Locally finite varieties with large free spectra

The free-spectrum and g-spectrum functions of a variety V are defined as fV(n) = “the size of the free algebra on n generators in V” and gV(n) = “the number of non-isomorphic algebras in V generated by n or fewer elements”, for any positive integer n. Thus V is locally finite iff for each positive integer n, fV(n) is a positive integer (and consequently gV(n) is a positive integer). If V is locally finite, then we have