We prove Khintchine type inequalities for words of a fixed length in a reduced free product of C∗-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length d is completely bounded with norm depending linearly on d. We then apply these results to various approximation properties ...

Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with א1 generators is represented. It w...

We characterize free poset algebras F (P ) over partially ordered sets and show that they can be represented by upper semi-lattice algebras. Hence, the uniqueness, in decomposition into normal form, using symmetric difference, of non-zero elements of F (P ) is established. Moreover, a characterization of upper semilattice algebras that are isomorphic to free poset algebras is given in terms of ...

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with “iterative algebras”, i. e., algeb...

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with “iterative algebras”, i. e., algeb...

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

For algebras A whose type is given by an endofunctor, iterativity means that every flat equation morphism in A has a unique solution. In our previous work we proved that every object generates a free iterative algebra, and we provided a coalgebraic construction of those free algebras. Iterativity w.r.t. an endofunctor was generalized by Tarmo Uustalu to iterativity w.r.t. a “base”, i.e., a func...

We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the C∗algebras and the non-sel...

We study finitely generated Heyting algebras from algebraic and model theoretic points of view. We prove amon others that finitely generated free Heyting algebras embed in their profinite completions, which are projective limits of finitely generated free Heyting algebras of finite dimension.

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...