Chang’s MV-algebras, on the one hand, are the algebras of the infinite-valued Łukasiewicz calculus and, on the other hand, are categorically equivalent to abelian lattice-ordered groups with a distinguished strong unit, for short, unital l-groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital l-groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras.

MV-algebras: a variety for magnitudes with archimedean units

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

/pdf/applying-universal-algebra-to-lambda-calculus-3wk3y30qvb.pdf

Applying Universal Algebra to Lambda Calculus

Review: W. A. Pogorzelski, Structural Completeness of the Propositional Calculus

We generalize to universal algebra concepts originating fromλ-calculus and programming to prove a new result onthe lattice λTof λ-theories, and ageneral theorem of pure universal algebra which can be seen as ameta version of the Stone Representation Theorem. In this paper weintroduce the class of Church algebras(which includes allBoolean algebras, combinatory algebras, rings with unit and theterm algebras of all λ-theories) to model the"if-then-else" instruction of programming. The interest of Churchalgebras is that each has a Boolean algebra of central elements,which play the role of the idempotent elements in rings. Centralelements are the key tool to represent any Church algebra as a weakBoolean product of indecomposable Church algebras and to prove themeta representation theorem mentioned above. We generalize thenotion of easy λ-term of lambda calculus to provethat any Church algebra with an "easy set" of cardinalitynadmits (at the top) a lattice interval of congruencesisomorphic to the free Boolean algebra with ngenerators.This theorem has the following consequence for the lattice ofλ-theories: for every recursively enumerableλ-theory φand each n, thereis a λ-theory φn? φsuch that {ψ: ψ? φn} "is" the Booleanlattice with 2 nelements.

/pdf/from-l-calculus-to-universal-algebra-and-back-b7kf122r0f.pdf

From λ-Calculus to Universal Algebra and Back

Almost structural completeness; an algebraic approach

Sheaf representation and Chinese remainder theorems

Central Elements in Varieties with the Fraser–Horn Property

Given a variety ν we study the existence of a class ℱ such that S1 every A e ν can be represented as a global subdirect product with factors in ℱ and S2 every non-trivial A e ℱ is globally indecomposable. We show that the following varieties (and its subvarieties) have a class ℱ satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.

Birkhoff-like sheaf representation for varieties of lattice expansions

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.

Semantical conditions for the definability of functions and relations

The Subquasivariety Lattice of a Discriminator Variety

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