Antonino Salibra

In recent work, the authors introduced the notion of n-dimensional Boolean algebra and the corresponding propositional logic nCL. In this paper, we introduce a sequent calculus for nCL and we show its soundness and completeness. Completeness relies on the generalisation to the n-ary case of the classical proof based on the Lindenbaum algebra of formulas and on Boolean ultrafilters.

We continue the investigation of Boolean-like algebras of dimension n (nBA) having n constants e1,...,en, and an (n+1)-ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBAs share many remarkable properties with the variety of Boolean algebras and with primal varieties. Exploiting the concept of central element, we extend the notion of Boolean power to that of semiring power and we prove two representation theorems: (i) Any pure nBA is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This gives a new proof of Foster's theorem on primal varieties.

We develop a new general framework for algebras and clones, called Universal Clone Algebra. Algebras and clones of finitary operations are to Universal Algebra what t-algebras and clone algebras are to Universal Clone Algebra. Clone algebras have been recently introduced to found a one-sorted, purely algebraic theory of clones, while t-algebras are first introduced in this article. We present a method to codify algebras and clones into t-algebras and clone algebras, respectively. We provide concrete examples showing that general results in Universal Clone Algebra, when translated in terms of algebras and clones, give more powerful versions of known theorems in Universal Algebra. We apply this methodology to Birkhoff's HSP theorem and to the recent topological versions of Birkhoff's theorem.

We introduce the notion of clone algebra, intended to found a one-sorted, purely algebraic theory of clones. Clone algebras are defined by true identities and thus form a variety in the sense of universal algebra. The most natural clone algebras, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras. The universe of a functional clone algebra, called omega-clone, is a set of infinitary operations containing the projections and closed under finitary compositions. We show that there exists a bijective correspondence between clones (of finitary operations) and a suitable subclass of functional clone algebras, called block algebras. Given a clone, the corresponding block algebra is obtained by extending the operations of the clone by countably many dummy arguments. One of the main results of this paper is the general representation theorem, where it is shown that every clone algebra is isomorphic to a functional clone algebra. In another result of the paper we prove that the variety of clone algebras is generated by the class of block algebras. This implies that every omega-clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to answer a classical question about the lattices of equational theories. The second application is to the study of the category VAR of all varieties. We introduce the category CA of all clone algebras (of arbitrary similarity type) with pure homomorphisms as arrows. We show that the category VAR is categorically isomorphic to a full subcategory of CA. We use this result to provide a generalisation of a classical theorem on independent varieties.

Skew Boolean algebras (SBA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed SBAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use SBAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed SBAs in terms of nBAs of n-partitions.

Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use skew BAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed skew BAs in terms of nBAs of n-partitions.