Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

Representations of algebras in varieties generated by infinite primal algebras

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\) ) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\) . Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\) ) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\) ) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\) ).     

Complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four

Using the theory of multidimensional three-webs we give a complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four.     

Strong regular varieties of partial algebras

Presented by H. P. Gumm.     

Reflection-closed varieties of multisorted algebras and minor identities

The notion of reflection is considered in the setting of multisorted algebras. The Galois connection induced by the satisfaction relation between multisorted algebras and minor identities provides a characterization of reflection-closed varieties: a variety of multisorted algebras is reflection-closed if and only if it is definable by minor identities. Minor-equational theories of multisorted algebras are described by explicit closure conditions. It is also observed that nontrivial varieties of multisorted algebras of a non-composable type are reflection-closed.     

Tolerances as images of congruences in varieties defined by linear identities

An identity s = t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra ${\mathcal{A}}$ in a variety defined by a set of linear identities. We prove that there exist an algebra ${\mathcal{B}}$ in the same variety and a congruence θ of ${\mathcal{B}}$ such that a homomorphism from ${\mathcal{B}}$ onto ${\mathcal{A}}$ maps θ onto T.     

On the regular part of varieties of algebras

We show that the regular part of a variety of algebras may not consist entirely of subalgebras of Plonka sums; indeed, the latter may not be closed under homomorphic images. In addition we give an example of a finitely based variety whose regular part has no finite basis (but is finitely based in a different similarity type).Our examples contrast with known results for semigroups, and contradict a published claim for the general case.To the memory of András HuhnPresented by Boris Schein.     

Decidability of elementary theories of finitely presented algebras of the variety of algebras defined by the empty system of identities

Translated from Matematicheskie Zametki, Vol. 45, No. 5, pp. 93–102, May, 1989.     

Representations,automorphisms, and derivations of some operator algebras

The representations of the algebra of bounded finite rank operators on a normed space are studied, and the results applied to related algebras. In particular, it is shown that every derivation of the algebra of all bounded operators is inner.     

Operator algebras with contractive approximate identities

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for C^?-algebras, and we give many other applications.