Diagonals and Block-Ordered Relations

HAL (Le Centre pour la Communication Scientifique Directe), 2016

2005

A topos version of Cantor’s back and forth theorem is established and used to prove that the ordered structure of the rational numbers 〈Q, <〉 is homogeneous in any topos with natural numbers object. The notion of effective homogeneity is introduced, and it is shown that 〈Q, <〉 is a minimal effectively homogeneous structure, that is, it can be embedded in every other effectively homogeneous ordered structure.

TACL 2009}, 2009

We study an interpretation of lattice connectives as natural join and inner union between database relations with non-uniform headers. To the best of our knowledge, this interpretation was proposed first by database researchers in [Tropashko, 2005, Spight and Tropashko, 2006]. It does not seem to have attracted the attention of algebraists even though connections between algebraic logic and relational databases have been discussed in the literature (see for example [Imielinski and Lipski, 1984] or [Düntsch and Mikulás, 2001]). ...

Annals of Mathematical Logic, 1980

introduced the notion of degree of unsolvability and the partial ordering ~ i on ~T, the set of such Tt, ring degrees, induced by Turing reducibility (Turing 13711, His paper with Kleene [ 1,4] contains the first serious analysis of this structure (~'r, ~x). They prove, for example, that all coantable partial orderings can be embedded in (ar, ~<~-). These embeddings show that the existential (it;st order~ theory of (~-r, ~r) is decidable, Next Spector [35], in a paper arising from Kleene's 1953 seminar, made an important inroad on the two quantifie, (i.e., VzI~ theory by showing that there is a minimal (Turing) degree. Sacks [31] extended these results and set forth some important conjectures on embeddings a~nd initial segments of ~'r. In particular he points out that one can prove the undecidability of the theory of (£ar, ~<v) by such results. This work inspired many papers by others eszabli,,hing better and better initial segment results. One milesto;~e was kachlan which showed that every countable distributive lattice can bc embedded as an initial segment of the Turing degrees. As the theory of distributive lattices was known to be undecidable, this sufficed to verify Sacks' conjezture that so is the theory of (@r, <~a-). (In fact it would have sufficed to embed all f-~nitc distributive lattices as was pointed out by Thomason [36] for hyperdegrees.) Two directions in which such results can be sharpened immediately come to mind, One is, where does the undecidability first arise in terms of quantifier complexity. The second is just how complicated is the full theory of (~, ~). (qhe results of Kleene and Post [ 14] showed only that the 3-theory was decidable while the coding of distributive lattices only showed that the full theory ha', degree at least 0'.) Further progress required further structural results. For the first question Lerman [20] supplied an essential ingredient by settling the full conjecture from Sacks , He showed that every finite lattice is embeddable ~, an initial segment of fib-. This can be combined with Kleene and Pc, st [14] to decide the 'q::l theory of

Studia Logica, 1993

The properties of antisymmetry and linearity axe easily seen to be sufficient for a recursively enumerable binary relation to be recursivdy isomorphic to a recursive relation. Removing either condition allows for the existence of a structure where no recursive isomorph exists, and natural examples of such structures axe surveyed.. This paper looks at recursively emtmerable binary relations, to explore conditions under which r.e. relations have recursive isomorphs. This question arises from several different perspectives. For unary relations there is a very simple answer: Every r.e. set is in 1-1 correspondence with a recursive set, and this correspondence itself cannot be required to be a recursive map because there are r.e. sets which are nonrecursive. For binary relations there are a number of well-known structures. It is known that every r.e. linear order is isomorphic to a recursive linear order, and that the isomorphism itself can be required to be recursive. But the same is not true for partial orders. Here there are r.e. models, including partial orders of fixed finite dimension, which are not isomorphic (recursively or otherwise) to recursive models. Another example is r.e. equivalence relations. The study of universal structures modulo r.e. equivalence relations has been very fruitful in the context of various lattices looked at from a recursive-algebraic viewpoint. (See Nerode and Remmel [4].) Here it is natural to ask if r.e. equivalence relations necessarily have recursive isomorphs. It turns out that very simple conditions on an r.e. binary relation guarantee that it will be recursively isomorphic to a recursive binary relation. Being antisymmetric and total (linear) suffices. This is of course the case for a linear order. What is surprising is that removing either condition allows for the existence of r.e. relations not isomorphic to recursive ones, and constructions from r.e. presented linear orders [7] and partial orders [8] are surveyed to establish that there are very natural examples of structures where this happens. Coding arguments suffice to show why these structures are inherently different from linear orders. In dosing, a result which codes arbitrary relations in terms of equivalence relations is cited and allows us to ask if these techniques can be extended to answer the question for equivalence relations. Notation is standard, as described in [6] or [9].

arXiv (Cornell University), 2023

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -specifically that all partial equivalence relations have an index-and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called "all or nothing" axiom used to facilitate pointwise reasoning is derivable from our axiom of choice.

Journal of Pure and Applied Algebra, 1987

The Prime Decomposition Theorem of Krohn and Rhodes for finite monoids emphasized the importance of division ('is a surmorphism of a submonoid of) for monoids. B. Tilson extended this concept to division of graphs and categories. In this strict Tilson ordering we prove that between every two finite connected non-trivial (there exists an object with a non-identity self arrow) categories or graphs there exists another such. * Supported in part by NSF grant DMS85-02367.

2025

The problem that will be the object of study throughout the development of this text can be summarized by presenting the following question: if everything is structure, and if structure is an upfold, and if the upfold is an ordered sequence of n elements, that is, a function whose domain is an ordered countable set, What does all this say about "reality"? The argument that follows basically consists of a long attempt to articulate a systematic and rigorous answer to this question.

Philosophia Mathematica, 2014

Many mathematical objects arise from equivalence classes, and invite implementation as those classes. Set existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories (e.g. NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for "low" sets, and thus-e.g.-a set of all (low) ordinals. However that set has an ordinal in turn, which is not a member of the set constructed, so no set of all ordinals is obtained thereby. This "recurrence problem" is discussed. This essay can be seen as a non-technical companion to [6] and is, like it, a precursor to-and will be subsumed into-the longer book-length treatment of these matters that I have promised myself I will write. 1

arXiv (Cornell University), 2014

Let C be a class of finite combinatorial structures. The profile of C is the function ϕ C which counts, for every integer n, the number ϕ C (n) of members of C defined on n elements, isomorphic structures been identified. The generating function of C is H C (x) := n 0 ϕ C (n)x n. Many results about the behavior of the function ϕ C have been obtained. Albert and Atkinson have shown that the generating series of several classes of permutations are algebraic. In this paper, we show how their results extend to classes of ordered binary relational structures; putting emphasis on the notion of hereditary well quasi order, we discuss some of their questions and answer one.