Day's Theorem is sharp for nn even

Both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from J\'onsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1}$ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, specular, mixed and defective. All the results hold also when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.Comment: v.2, some improvements and some corrections, particularly in Section 9 v.3, a few further improvements, corrections simplification

Classical Opacity

Philosophy and Phenomenological Research, EarlyView

The distributivity spectrum of Baker's variety

For every $n$, we evaluate the smallest $k$ such that the congruence inclusion $\alpha (\beta \circ_n \gamma ) \subseteq \alpha \beta \circ_{k} \alpha \gamma$ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.Comment: v3, v4 improvements and simplifications v5, v6 minor fixe

Leibniz's Principles and Topological Extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine the real things". Here we give a precise formulation of these principles within the framework of the Topological Extensions of [8], structures that generalize at once compactifications, completions, and nonstandard extensions. In this topological context, the above Leibniz's principles appear as a property of separation, a property of compactness, and a property of analyticity, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnz's principles can be fulfilled in pairs, but not all three together.Comment: 16 page

The Lanczos potential for Weyl-candidate tensors exists only in four dimensions

We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W_abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L_abc.Comment: 11 pages, LaTeX, Heavily revised April 200

On uniqueness of end sums and 1-handles at infinity

For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. The present paper examines how and when uniqueness fails. Examples are given, in the categories TOP, PL and DIFF, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infinite) collections of 0- and 1-handles at infinity. Various applications are presented, including an analysis of how the monoid of smooth manifolds homeomorphic to R^4 acts on the smoothings of any noncompact 4-manifold.Comment: 25 pages, 8 figures. v2: Minor expository improvement

A topological interpretation of three Leibnizian principles within the functional extensions

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object" (Identity of indiscernibles), "everything can possibly exist, unless it yields contradiction" (Possibility as consistency), and "the ideal elements correctly determine the real things" (Transfer). Here we give a precise logico-mathematical formulation of these principles within the framework of the Functional Extensions, mathematical structures that generalize at once compactifications, completions, and elementary extensions of models. In this context, the above Leibnizian principles appear as topological or algebraic properties, namely: a property of separation, a property of compactness, and a property of directeness, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnizian principles may be fulfilled in pairs, but not all three together.Comment: arXiv admin note: substantial text overlap with arXiv:1012.434