Representability and local representability of algebraic theories

A finitary monosorted algebraic theory is called locally representable (or representable) in a category with finite products if every its initial segment is the domain of a full faithful finite-products-preserving functor into (or if itself is the domain of such a functor). The question of when local representability implies representability is discussed, and theories for which local representability always implies representability are fully characterized. Received December 20, 1996; accepted in final form March 19, 1998.     

Algebras with a compatible uniformity

Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a generalization of the lattice of congruences Con A. Mal'cev properties of V influence the structure of Unif A, much as they do that of Con A. The category V[CHUnif] of complete, Hausdor. such algebras in the variety V is particularly interesting; it has a factorization system , and V embeds into V[CHUnif] in such a way that is the subcategory of onto and the subcategory of one-one homomorphisms. Received February 17, 2000; accepted in final form April 1, 2001.     

On locally finite varieties with undecidable equational theory

No Abstract. Received July 8, 1999; accepted in final form January 31, 2001.     

On order-polynomial completeness of lattices

In this note we prove that the cardinality of an infinite order-polynomially complete lattice (if such a lattice exists) must be greater than each , where is defined such that and for . This strengthens the result in [4]. Received November 10, 1995; accepted in final form April 6, 1998.     

Some universal properties of the category of clones

In [6], O. C. García and W. Taylor asked if the breadth of the lattice of interpretability types of varieties is uncountable. The present paper solves the problem by two different constructions. Both of them show that any cardinal number is the cardinality of an antichain in the named lattice and that the existence of a proper class antichain is equivalent to the negation of Vopěnka's principle. The first construction gives in a way a minimal solution of the problem, whereas the second one gives stronger results about the category of clones. Received November 12, 1999; accepted in final form June 19, 2001.     

Algebraic coalitions

The idea of an algebra in is introduced. Within congruence modular varieties such algebras are shown to be the abelian algebras with a one-element subalgebra. This leads on to the notion of algebraic coalition, which is characterized for congruence modular varieties and for varieties of Jónsson–Tarski algebras. This characterization displays an intimate relationship between algebraic coalitions, Gumm difference terms, and the centre of an algebra. Received July 16, 1996; accepted in final form May 2, 1997.     

Algebraic coalitions: 2

This paper continues the investigation of [2] into the notions of algebra in and of coalition. Here these notions are characterized for the variety of semilattices, more generally for any variety of inverse semigroups, and for subtractive varieties. Received June 25, 1997 accepted in final form October 16, 1998.     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Maltsev conditions and relations on algebras

We show that every sentence preserved by products in a purely relational first order language corresponds to a Maltsev condition on subalgebras of direct powers. Moreover, we establish that this correspondence captures all strong Maltsev conditions whose defining equations do not involve compositions. We then demonstrate how a broader version of the correspondence is sufficient to capture all Maltsev conditions when we restrict our attention to locally finite varieties. Received November 6, 1998; accepted in final form July 8, 1999.     

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 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 , where is an n-cycle of degree n 2. Interpretability types of the cyclic varieties form, in , a subsemilattice isomorphic to a semilattice of square-free natural numbers n 2, under taking m n=[m,n] (l.c.m.).     

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice. Received February 9, 2000; accepted in final form November 23, 2000.     

Discriminator or dual discriminator?

Varieties are considered with p(x, y, z), a single ternary operation, which acts as a local discriminator or dual discriminator on the subdirectly irreducible elements. If p(x, y, z) is "global", then all subvarieties are finitely based. In the general case a continuum of non-finitely based subvarieties are presented. A graph theoretical picture leads to a variety of groupoids connecting the left-zero and the right-zero semigroups. For this variety some open problems are presented. Received October 7, 1998; accepted in final form October 4, 1999.     

On equational theories of varieties of anticommutative rings

The existence of a finitely based variety of anticommutative rings (in the sense of the identityx ²=0) with unsolvable equational theory is proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 230–245, February, 1999.     

Homotopy Theories of Algebras over Operads

Homotopy theories of algebras over operads, including operads over “little n-cubes,” are defined. Spectral sequences are constructed and the corresponding homotopy groups are calculated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 278–285.Original Russian Text Copyright © 2005 by V. A. Smirnov.     

Dynamic programming for some optimal control problems with hysteresis

We study an infinite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary differential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) first order Hamilton-Jacobi equation, which in the first case is of finite dimension and in the second case is of infinite dimension. In both cases we prove that the value function is the only bounded uniformly continuous viscosity solution of the equation.     

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

Fulton and MacPherson asked if there exists a bivariant version of the Chern-Schwartz-MacPherson class. Brasselet solved this problem affirmatively in the category of analytic varieties and cellular morphisms. However, it has not been solved in the general case and the uniqueness of such a bivariant Chern class is still open. In this paper we show the unique existence of the bivariant Chern-Schwartz-MacPherson class with values in Chow groups. To be more precise, we show that there exists a unique Grothendieck transformation from the bivariant theory of constructible functions to Fulton-MacPherson's operational bivariant theory of Chow groups, provided that the compatibility with flat pullback is not required on the operational bivariant theory.