Coalgebras on Measurable Spaces

Thesis (PhD) - Indiana University, Mathematics, 2005Given an endofunctor T in a category C, a coalgebra is a pair (X,c) consisting of an object X and a morphism c:X ->T(X). X is called the carrier and the morphism c is called the structure map of the T-coalgebra. The theory of coalgebras has been found to abstract common features of different areas like computer program semantics, modal logic, automata, non-well-founded sets, etc. Most of the work on concrete examples, however, has been limited to the category Set. The work developed in this dissertation is concerned with the category Meas of measurable spaces and measurable functions. Coalgebras of measurable spaces are of interest as a formalization of Markov Chains and can also be used to model probabilistic reasoning. We discuss some general facts related to the most interesting functor in Meas, Delta, that assigns to each measurable space, the space of all probability measures on it. We show that this functor does not preserve weak pullbacks or omega op-limits, conditions assumed in many theorems about coalgebras. The main result will be two constructions of final coalgebras for many interesting functors in Meas. The first construction (joint work with L. Moss), is based on a modal language that lets us build formulas that describe the elements of the final coalgebra. The second method makes use of a subset of the projective limit of the final sequence for the functor in question. That is, the sequence 1 <- T1 <- T 2 1 <-... obtained by iteratively applying the functor to the terminal element 1 of the category. Since these methods seem to be new, we also show how to use them in the category Set, where they provide some insight on how the structure map of the final coalgebra works. We show as an application how to construct universal Type Spaces, an object of interest in Game Theory and Economics. We also compare our method with previously existing constructions

Superrational types

We present a formal analysis of Douglas Hofstadter?s concept of superrationality. We start by defining superrationally justifiable actions, and study them in symmetric games. We then model the beliefs of the players, in a way that leads them to different choices than the usual assumption of rationality by restricting the range of conceivable choices. These beliefs are captured in the formal notion of type drawn from epistemic game theory. The theory of coalgebras is used to frame type spaces and to account for the existence of some of them. We find conditions that guarantee superrational outcomes.Fil: Tohmé, Fernando Abel. 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: Viglizzo, Ignacio Dario. 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; Argentin

Structural relations of symmetry among players in strategic games

The notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strategic games.Fil: Tohmé, Fernando Abel. 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; Argentina. Universidad Nacional del Sur. Departamento de Economía; ArgentinaFil: Viglizzo, Ignacio Dario. 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; Argentin

Construction of Nelson Algebras

The Vakarelov construction of Nelson algebras up from Heyting ones is generalized to obtain De Morgan algebras from distributive lattices. Necessary and sufficient conditions for these De Morgan algebras to be Nelson algebras are shown, and a characterization of the join-irreducible elements in the finite case is given.Fil: Monteiro, Luiz F.. Universidad Nacional del Sur; ArgentinaFil: Viglizzo, Ignacio Dario. 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; ArgentinaXIV Congreso Dr. Antonio A. R. MonteiroBahía BlancaArgentinaUniversidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática de Bahía Blanc

Free monadic Tarski and MMI3-algebras

MMI3-algebras are a generalization of the monadic Tarski algebras as defined by A. Monteiro and L. Iturrioz, and a particular case of the MMIn+1-algebras defined by A. Figallo. They can also be seen as monadic three-valued £ukasiewicz algebras without a first element. By using this point of view, and the free monadic extensions, we construct the free MMI3-algebras on a finite number of generators, and indicate the coordinates of the generators. As a byproduct, we also obtain a construction of the free monadic Tarski algebras.Fil: Entizne, Rosana V.. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Monteiro, Luiz F.. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Savini, Sonia M.. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Viglizzo, Ignacio Dario. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentin

A coalgebraic approach to type spaces

When two or more players are engaged in a game with uncertainties, they need to consider what the other players’ beliefs may be, which in turn are influenced by what they think the first player’s ideas are. Harsanyi defined type spaces simply as a set in which all possible players-as defined by their beliefs- could be found. Later on, more meaningful constructions of this set were performed. The theory of coalgebra, on the other hand, has been created to deal with circular phenomena, so its application to the problem of type spaces is only natural. We show how to apply it and we use the more general framework of category theory to compare the relative strength of previous solutions to the problem of defining type spaces.Fil: Viglizzo, Ignacio Dario. 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; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Universidad Nacional del Sur. Departamento de Ciencias e Ingeniería de la Computación. Instituto de Ciencias e Ingeniería de la Computación; Argentin

On Some Semi-Intuitionistic Logics

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics.Fil: Cornejo, Juan Manuel. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; ArgentinaFil: Viglizzo, Ignacio Dario. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentin

Semi-Nelson Algebras

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.Fil: Cornejo, Juan Manuel. 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: Viglizzo, Ignacio Dario. 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; Argentin

Semi-Intuitionistic Logic with Strong Negation

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras.Fil: Cornejo, Juan Manuel. 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: Viglizzo, Ignacio Dario. 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; ArgentinaXIV Congreso Dr. Antonio MonteiroBahía BlancaArgentinaUniversidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática de Bahía Blanc

Semi-intuitionistic Logic with Strong Negation

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras.Fil: Cornejo, Juan Manuel. 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: Viglizzo, Ignacio Dario. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin