One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pair...

0 Introduction For unital C-algebras endowed with states there is a natural reduced free product construction which generalizes the C-algebra of the regular representation of a free product group. Whereas the question of computing the K-theory and all the various KK-groups is completly understood in the case of discrete groups (see the work of Pimsner in 13]), little is known so far for more ge...

Most of the studies on Rota–Baxter algebras (also known as Baxter algebras) have been for commutative algebras. Free commutative Rota–Baxter algebras were constructed by Rota and Cartier in the 1970s. A later construction was obtained by Keigher and one of the authors in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the...

Most of the studies on Rota–Baxter algebras (also known as Baxter algebras) have been for commutative algebras. Free commutative Rota–Baxter algebras were constructed by Rota and Cartier in the 1970s. A later construction was obtained by Keigher and one of the authors in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the...

We show that for finite n ≥ 3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived fr...

Completely iterative theories of Calvin Elgot formalize (potentially infinite) computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is that infinite trees form a free completely iterative theory. Their algebraic proof of this result is extremely complicated. We present completely iterative algebras as a new approach to the description of free comp...

In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and nonunitary Rota-Baxter algebras, in terms the method Gr\"{o}bner-Shirshov bases.