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On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
Linear Space Algorithm for Online Detection of Global Predicates
 PROC. INTERNATIONAL WORKSHOP ON STRUCTURES IN CONCURRENCY THEORY (STRICT '95
, 1995
"... A fundamental problem in debugging and monitoring is detecting whether the state of a system satisfies some predicate. Cooper and Marzullo defined this problem as Possibly(\Phi) for distributed computations. This paper presents the first online algorithm using linear space which resolve this p ..."
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Cited by 15 (0 self)
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A fundamental problem in debugging and monitoring is detecting whether the state of a system satisfies some predicate. Cooper and Marzullo defined this problem as Possibly(\Phi) for distributed computations. This paper presents the first online algorithm using linear space which resolve this problem in the general case, improving all existing algorithms both in time and space. It is particularly interesting for the detection of Possibly(\Phi) on potentially infinite computations. To our knowledge, it also the only algorithm of detection which do not make use of vectors of timestamps. The presented algorithm is based on a structural properties of the consistent cuts lattice, leading to a new structure to study distributed computations: the consistent cuts tree.
Computing OnLine the Lattice of Maximal Antichains of Posets
, 1994
"... This paper is dedicated to the online computation of the lattice of maximal antichains, say g MA(P ) = (MA(P ); MA(P ) ), of a finite poset e P = (P; P ). This online computation fulfills what we call the linear extension hypothesis, that is when the ordering of the incoming vertices of e P fol ..."
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Cited by 14 (1 self)
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This paper is dedicated to the online computation of the lattice of maximal antichains, say g MA(P ) = (MA(P ); MA(P ) ), of a finite poset e P = (P; P ). This online computation fulfills what we call the linear extension hypothesis, that is when the ordering of the incoming vertices of e P follows a linear extension of e P . Beside its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behaviours. Particularly this lattice of maximal antichains seems interesting for testing distributed computations, domain where the lattice of antichains is going to be widely used. The online algorithm we propose for this computation has a run time complexity in O((jP j +! 2 (P ))!(P )jMA(P )j) which strictly improves the previous offline algorithms.
EQUIVALENCE BETWEEN FRAÏSSÉ’S CONJECTURE AND JULLIEN’S THEOREM.
, 2004
"... We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a th ..."
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Cited by 7 (4 self)
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We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddablity, contains no infinite descending chain and no infinite antichain. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of RCA0+Σ1 1IND. We also prove that Fraïssé’s conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+, −}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings. While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just ATR0+Σ1 1IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, which does not embed L has a linearization, hyperarithmetic (or equivalently ∆1 1) in P ⊕ L, which does not embed L.
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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Cited by 4 (1 self)
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
BEYOND THE ARITHMETIC
, 2005
"... Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees ..."
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Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees, of course, preserving jump and join; we show that every finite partial ordering labeled with the classes in the generalized high/low hierarchy can be embedded into the Turing degrees; we show that every generalized high degree has the complementation property; and we show that if a Turing degree a is either 1generic and ∆ 0 1, 2generic and arithmetic, nREA, or arithmetically generic, then the theory of the partial ordering of the Turing degrees below a is recursively isomorphic to true first order arithmetic. Second, we work with equimorphism types of linear orderings from the viewpoints of Computable Mathematics and Reverse Mathematics. (Two linear orderings are equimorphic if they can be embedded in each other.) Spector proved in 1955 that every hyperarithmetic ordinal is isomorphic to a recursive one. We extend his result and prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. From the viewpoint of Reverse Mathematics, we look at the strength of Fraïssé’s conjecture. From our results, we deduce that Fraïssé’s conjecture is sufficient and necessary to develop a reasonable theory of equimorphism types of linear orderings. Other topics we include in this thesis are the following: we look at structures for which Arithmetic Transfinite Recursion is the natural system to study them; we study theories of hyperarithmetic analysis and present a new natural example; we look at the complexity of the elementary equivalence problem for Boolean algebras; and we prove that there is a minimal pair of Kolmogorovdegrees. BIOGRAPHICAL SKETCH I was born and raised in Montevideo, Uruguay. I started to take mathematics
An overview of Delta type operations on quasisymmetric functions
"... We present an overview oftype operations on the algebra of quasisymmetric functions. Nous pr'esentons un survol de l'ensemble des propri'et'es de typeanneau de l'alg`ebre des fonctions quasisym'etriques. 1 ..."
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We present an overview oftype operations on the algebra of quasisymmetric functions. Nous pr'esentons un survol de l'ensemble des propri'et'es de typeanneau de l'alg`ebre des fonctions quasisym'etriques. 1
ON SCATTERED POSETS WITH FINITE DIMENSION
, 812
"... Abstract. We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. Introduction and presentation of the results A fundamental result, due to Szpilrajn [26], states that every order on a set is the int ..."
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Abstract. We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. Introduction and presentation of the results A fundamental result, due to Szpilrajn [26], states that every order on a set is the intersection of a family of linear orders on this set. The dimension of the order, also called the dimension of the ordered set, is then defined as the minimum cardinality of such a family (Dushnik, Miller [11]). Specialization of Szpilrajn’s result to several
THE LENGTH OF CHAINS IN ALGEBRAIC LATTICES
, 812
"... Abstract. We study how the existence in an algebraic lattice L of a chain of a given type is reflected in the joinsemilattice K(L) of its compact elements. We show that for every chain α of size κ, there is a set B of at most 2 κ joinsemilattices, each one having a least element such that an algeb ..."
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Abstract. We study how the existence in an algebraic lattice L of a chain of a given type is reflected in the joinsemilattice K(L) of its compact elements. We show that for every chain α of size κ, there is a set B of at most 2 κ joinsemilattices, each one having a least element such that an algebraic lattice L contains no chain of order type I(α) if and only if the joinsemilattice K(L) of its compact elements contains no joinsubsemilattice isomorphic to a member of B. We show that among the joinsubsemilattices of [ω] <ω belonging to B, one is embeddable in all the others. We conjecture that if α is countable, there is a finite set B.