About Quotient Orders and Ordering Sequences

SummaryIn preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σf : D → ℝ, Σf (X)= ∑x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article can then be formulated as: ∑x∈Af(x)=∑X∈DΣf(X)(=∑X∈D∑x∈Xf(x)) After that (weakly) ascending/descending finite sequences (based on [3]) are introduced, in analogous notation to their infinite counterparts introduced in [18] and [13].The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from [16], where finite sequence of real numbers were sorted.The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. [10]).Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences s, t on a set X, such that rng t ⊆ rng s, and a function f : X → ℝ such that f is zero for every x ∈ rng s \ rng t, we have ∑ f o s = ∑ f o t.Johannes Gutenberg University, Mainz, GermanyGrzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest. Introduction to algorithms. MIT Press, 3. ed. edition, 2009. ISBN 0-262-53305-7, 978-0-262-53305-8, 978-0-262-03384-8. http://scans.hebis.de/HEBCGI/show.pl?21502893_toc.pdf.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.Gilbert Lee and Piotr Rudnicki. Dickson’s lemma. Formalized Mathematics, 10(1):29–37, 2002.Michael Maschler, Eilon Solan, and Shmuel Zamir. Game theory. Cambridge Univ. Press, 2013. ISBN 978-1-107-00548-8. doi: 10.1017/CBO9780511794216.Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179–184, 2001.Yatsuka Nakamura. Sorting operators for finite sequences. Formalized Mathematics, 12 (1):1–4, 2004.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Formalized Mathematics, 6(3):339–343, 1997.Bernd S. W. Schröder. Ordered Sets: An Introduction. Birkhäuser Boston, 2003. ISBN 978-1-4612-6591-7. https://books.google.de/books?id=hg8GCAAAQBAJ.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.25212113

A Homological Approach to Factorization

Mott noted a one-to-one correspondence between saturated multiplicatively closed subsets of a domain D and directed convex subgroups of the group of divisibility D. With this, we construct a functor between inclusions into saturated localizations of D and projections onto partially ordered quotient groups of G(D). We use this functor to construct many cochain complexes of o-homomorphisms of po-groups. These complexes naturally lead to some fundamental structure theorems and some natural homology theory that provide insight into the factorization behavior of D.Comment: Submitted for publication 12/15/201

A scattering of orders

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $\mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $\mathcal B$. More generally, we say that a partial ordering is $\kappa$-scattered if it does not contain a copy of any $\kappa$-dense linear ordering. We prove analogues of Hausdorff's result for $\kappa$-scattered linear orderings, and for $\kappa$-scattered partial orderings satisfying the finite antichain condition. We also study the $\mathbb{Q}_\kappa$-scattered partial orderings, where $\mathbb{Q}_\kappa$ is the saturated linear ordering of cardinality $\kappa$, and a partial ordering is $\mathbb{Q}_\kappa$-scattered when it embeds no copy of $\mathbb{Q}_\kappa$. We classify the $\mathbb{Q}_\kappa$-scattered partial orderings with the finite antichain condition relative to the $\mathbb{Q}_\kappa$-scattered linear orderings. We show that in general the property of being a $\mathbb{Q}_\kappa$-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions

Alternating normal forms for braids and locally Garside monoids

We describe new types of normal forms for braid monoids, Artin-Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties (``locally Garside monoids''). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard ordering (``Dehornoy order'') of Bn in terms of that of B(n-1), and to a quadratic upper bound for the complexity of this ordering

Modelling Concurrency with Comtraces and Generalized Comtraces

Comtraces (combined traces) are extensions of Mazurkiewicz traces that can model the "not later than" relationship. In this paper, we first introduce the novel notion of generalized comtraces, extensions of comtraces that can additionally model the "non-simultaneously" relationship. Then we study some basic algebraic properties and canonical reprentations of comtraces and generalized comtraces. Finally we analyze the relationship between generalized comtraces and generalized stratified order structures. The major technical contribution of this paper is a proof showing that generalized comtraces can be represented by generalized stratified order structures.Comment: 49 page