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Arturo De Faveri

Publications and source records attributed to Arturo De Faveri.

4 recordsLinked to original sources
arXiv2026

What is a Model of the Linear Lambda Calculus?

We investigate the notion of model of the linear $λ$-calculus from an algebraic perspective. Our starting point is the operad of linear $λ$-terms, whose algebras provide a natural candidate. We prove that this notion of model is equivalent to two other structures: a linear analogue of Curry's $λ$-algebras, and semiclosed operads, a class of operads equipped with an internal abstraction operation. The equivalence between these three approaches unifies three complementary answers to the question of what should be regarded as a model of the linear $λ$-calculus. As a second contribution, we give a finite equational presentation for the linear variant of $λ$-algebras using the linear combinators $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{I}$. Finally, exploiting the equivalence with semiclosed operads, we establish a linear analogue of Scott's representation theorem by showing that every model arises as a reflexive object in a natural monoidal closed category of presheaves.

cs.LO↗
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↗
arXiv2026

Rings and Boolean Algebras as Algebraic Theories

We present a unified framework for representing commutative rings through affine algebraic theories and Boolean rings through hyperaffine algebraic theories. This yields categorical equivalences between these theories and, respectively, commutative rings and Boolean rings. We then analyse models of affine theories over a Boolean ring $B$, comparing them with the models of hyperaffine theories, the well-known $B$-sets. Two novel characterisations are presented: the first defines these models as Boolean vector spaces equipped with an action of the Boolean ring; the second provides a representation in terms of sheaves, in analogy with $B$-sets. Finally, we establish a connection between hyperaffine theories and multidimensional Boolean algebras, a recently introduced generalisation of Boolean algebras.

math.LO↗