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Antonino Salibra

Publications and source records attributed to Antonino Salibra.

12 recordsLinked to original sources
arXiv2026

From clones to cm-monoids

Clones of functions play a foundational role in both universal algebra and theoretical computer science. In this work, we introduce clone merge monoids (cm-monoids), a unifying one-sorted algebraic framework that integrates abstract clones, clone algebras (previously introduced by the first and the third author), and Neumann's aleph0-abstract clones, while modelling the interplay of infinitary operations. Cm-monoids combine a monoid structure with a new algebraic structure called merge algebra, capturing essential properties of infinite sequences of operations. We establish a categorical equivalence between clone algebras and finitely-ranked cm-monoids. This equivalence yields by restriction a three-fold equivalence between abstract clones, finite-dimensional clone algebras, and finite-dimensional, finitely ranked cm-monoids, and is itself obtained by restriction from a categorical equivalence between partial infinitary clone algebras (which generalise clone algebras) and extensional cm-monoids. In a companion work, we develop the theory of modules over cm-monoids, offering a unified approach to polymorphisms and invariant relations, in the hope of providing new insights into algebraic structures and CSP complexity theory.

math.CT↗
arXiv2026

Groups and Inverse Semigroups in Lambda Calculus

We study invertibility of $λ$-terms modulo $λ$-theories. Here a fundamental role is played by a class of $λ$-terms called finite hereditary permutations (FHP) and by their infinite generalisations (HP). More precisely, FHPs are the invertible elements in the least extensional $λ$-theory $λη$ and HPs are those in the greatest sensible $λ$-theory $H^*$. Our approach is based on inverse semigroups, algebraic structures that generalise groups and semilattices. We show that FHP modulo a $λ$-theory $T$ is always an inverse semigroup and that HP modulo $T$ is an inverse semigroup whenever $T$ contains the theory of Böhm trees. An inverse semigroup comes equipped with a natural order. We prove that the natural order corresponds to $η$-expansion in $\mathrm{FHP} /T$, and to infinite $η$-expansion in $\mathrm{HP}/T$. Building on these correspondences we obtain the two main contributions of this work: firstly, we recast in a broader framework the results cited at the beginning; secondly, we prove that the FHPs are the invertible $λ$-terms in all the $λ$-theories lying between $λη$ and $H^+$. The latter is Morris' observational $λ$-theory, defined by using the $β$-normal forms as observables.

cs.LO↗
arXiv2024

Exploring New Topologies for the Theory of Clones

Clones of operations of arity omega (referred to as omega-operations) have been employed by Neumann to represent varieties of infinitary algebras defined by operations of at most arity omega. More recently, clone algebras have been introduced to study clones of functions, including omega-operations, within the framework of one-sorted universal algebra. Additionally, polymorphisms of arity omega, which are omega-operations preserving the relations of a given first-order structure, have recently been used to establish model theory results with applications in the field of complexity of CSP problems. In this paper, we undertake a topological and algebraic study of polymorphisms of arity omega and their corresponding invariant relations. Given a set A and a Boolean ideal X on the set of omega-sequences of elements of A, we propose a method to endow the set of omega-operations on A with a topology, which we refer to as X-topology. Notably, the topology of pointwise convergence can be retrieved as a special case of this approach. Polymorphisms and invariant relations are then defined parametrically, with respect to the X-topology. We characterise the X-closed clones of omega-operations in terms of polymorphisms and invariant relations of arity omega, and present a method to relate those infinitary invariant relation and polymorphisms to the classical (finitary) Inv-Pol.

cs.LO↗
arXiv2024

The higher dimensional propositional calculus

In recent research, some of the present authors introduced the concept of an n-dimensional Boolean algebra and its corresponding propositional logic nCL, generalising the Boolean propositional calculus to n>= 2 perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for nCL, named nLK. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that nLK enjoys the cut admissibility property. The latter relies on the generalisation to the n-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters.

cs.LO↗
arXiv2022

Boolean-like algebras of finite dimension

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.

cs.LO↗
arXiv2022

Universal Clone Algebra

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.

math.LO↗
arXiv2021

An algebraic theory of clones with an application to a question of Birkhoff and Maltsev

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.

math.LO↗
arXiv2019

On noncommutative generalisations of Boolean algebras

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.

math.LO↗
arXiv2013

Ordered Models of the Lambda Calculus

Answering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity (namely n-subtractivity) for partially ordered algebras. Finally we study the relation between n-subtractivity and relativized separation conditions in topological algebras, obtaining an incompleteness theorem for a general topological semantics of lambda calculus.

cs.LO↗
arXiv2013

The stack calculus

We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of expressions.

cs.LO↗
arXiv2013

Minimal lambda-theories by ultraproducts

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the least lambda-theory lambda-beta or the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper, we give a general tool to answer positively to this question and we apply it to a wide class of webbed models: the i-models. The method then applies also to graph models, Krivine models, coherent models and filter models. In particular, we build an i-model whose theory is the set of equations satisfied in all i-models.

cs.LO↗