Variations on Algebra: monadicity and generalisations of equational theories

Dedicated to Rod Burstal

Exploring the Boundaries of Monad Tensorability on Set

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We present two main results: we show that the tensor of two monads need not in general exist by presenting two counterexamples, one of them involving finite powerset (i.e. the theory of join semilattices); this solves a somewhat long-standing open problem, and contrasts with recent results that had ruled out previously expected counterexamples. On the other hand, we show that tensors with bounded powerset monads do exist from countable powerset upwards

The k-variable property is stronger than H-dimension k

Accepted versio

The First-Order Theory of Sets with Cardinality Constraints is Decidable

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is undecidable. Our language allows relating the cardinalities of sets to the values of integer variables, and can distinguish finite and infinite sets. We use quantifier elimination to show the decidability and obtain an elementary upper bound on the complexity. Precise program analyses can use our decidability result to verify representation invariants of data structures that use an integer field to represent the number of stored elements.Comment: 18 page

Quantum monadic algebras

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras and subfactors. Here we develop the basic properties of these quantum monadic and cylindric algebras and relate them to quantum predicate logic

Monadic Wajsberg hoops

Wajsberg hoops are the { , →, 1}-subreducts (hoop-subreducts) of Wajsberg algebras, which are term equivalent to MV-algebras and are the algebraic models of Lukasiewicz infinite-valued logic. Monadic MV-algebras were introduced by Rutledge [Ph.D. thesis, Cornell University, 1959] as an algebraic model for the monadic predicate calculus of Lukasiewicz infinitevalued logic, in which only a single individual variable occurs. In this paper we study the class of { , →, ∀, 1}-subreducts (monadic hoop-subreducts) of monadic MV-algebras. We prove that this class, denoted by MWH, is an equational class and we give the identities that define it. An algebra in MWH is called a monadic Wajsberg hoop. We characterize the subdirectly irreducible members in MWH and the congruences by monadic filters. We prove that MWH is generated by its finite members. Then, we introduce the notion of width of a monadic Wajsberg hoop and study some of the subvarieties of monadic Wajsberg hoops of finite width k. Finally, we describe a monadic Wajsberg hoop as a monadic maximal filter within a certain monadic MValgebra such that the quotient is the two element chain.Fil: Díaz Varela, José Patricio. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Cimadamore, Cecilia Rossana. Universidad Nacional del Sur. Departamento de Matemática; Argentin