Minimal Varieties and Identities of Relatively Free Algebras

Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient A of a given graded algebra R/c, whose Hilbert function is maximal among such quotients with the given socle degrees. For us c is usually a “general” complete intersection and we usually require that A be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that f...

Free Algebras in Varieties

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility. In universal algebra we deal with varieties – classes o...

Minimal Varieties of Algebras of Exponential Growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of a given exponent and of finite basic rank. As a consequence we describe the corresponding T-ideals of the free algebra, and we compute the asymptotics of the related codimension sequences. We then verify in this sett...

Relatively Free Algebras with Weak Exchange Properties

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and finitely gen...

Polynomial identities of some minimal varieties with low PI - exponent