Complete Partial Orders Research Articles

Some progress on the Aharoni–Korman conjecture

Aharoni and Korman (1992) [3] conjectured that every ordered set with no infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain.The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for (3+1)-free posets with no infinite antichains.We give a necessary and sufficient condition for a P4-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite P4-free graphs are finite cographs. We also prove that N-free chain complete posets and N-free posets with no infinite antichains are series-parallel.As a consequence, we obtain that the conjecture is true for N-free posets with no infinite antichain.

Poset Structure Concerning Cylindric Diagrams

Cylindric diagrams admit the structure of infinite $d$-complete posets with natural ordering. The purpose of this paper is to provide a realization of a cylindric diagram as a subset of an affine root system of type A via colored hook lengths and to present several characterizations of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the affine Weyl group.

Apartness, sharp elements, and the Scott topology of domains

AbstractWorking constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets.

Not every countable complete distributive lattice is sober

AbstractThe study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $\Sigma P\times \Sigma Q$ coincides with the Scott topology of the product poset $P\times Q$ if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and $\Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.

Injective and projective poset acts

In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {\bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {\bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {\bf PosAct}-$S$. Finally, we study injectivity and projectivity in {\bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {\bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.

On Small Types in Univalent Foundations

We investigate predicative aspects of constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky's propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. Moreover, we prove that locally small, nontrivial (directed or bounded) complete posets necessarily lack decidable equality. We prove our results for a general class of posets, which includes e.g. directed complete posets, bounded complete posets, sup-lattices and frames. Secondly, the fact that these nontrivial posets are necessarily large has the important consequence that Tarski's theorem (and similar results) cannot be applied in nontrivial instances. Furthermore, we explain that generalizations of Tarski's theorem that allow for large structures are provably false by showing that the ordinal of ordinals in a univalent universe has small suprema in the presence of set quotients. The latter also leads us to investigate the inter-definability and interaction of type universes of propositional truncations and set quotients, as well as a set replacement principle. Thirdly, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families.

A topological duality for dcpos

Abstract In this paper, a Stone-type duality for directed complete posets with a top element is developed by using a class of special subsets, named prime Scott open subsets. Following this idea, a topological duality for complete lattices is also obtained.

Algebraicness properties on transitive binary relational sets and a characterization of continuous directed complete posets

In this work we introduce and study the concepts of algebraicness, and continuity on transitive binary relational sets ( so called T RS). Some interactions between these concepts are investigated. Further more a characterization of continuous directed complete posets and Algebraic T RS are studied. Our results are generalizations of corresponding results in posets.

Grounding Game Semantics in Categorical Algebra

I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.

Transporting continuity properties from a poset to its subposets

We identify two key conditions that a subset A of a poset P may satisfy to guarantee the transfer of continuity properties from P to A. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset P whose way-below relation is the restriction of the way-below relation of P, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system.

Sharp Elements and Apartness in Domains

Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vita apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals and the lower reals.

Interior Operators Generated by Ideals in Complete Domains

This article presents some properties of a special class of interior operators generated by ideals. The mathematical framework is given by complete domains, namely complete posets in which the set of minimal elements is a basis. The first part of the paper presents some preliminary results; in the second part we present the novel interior operator denoted by G(i,I), an operator built starting from an interior operator i and an ideal I. Various properties of this operator are presented; in particular, the connection between the properties of the ideal I and the properties of the operator G(i,I). Two such properties (denoted by Pi and Qi) are extensively analyzed and characterized. Additionally, some characterizations for compact elements are presented.

Classifications of $\Gamma$-Colored $d$-Complete Posets and Upper $P$-Minuscule Borel Representations

The $\Gamma$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $\Gamma$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $\Gamma$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $\Gamma$-colored $d$-complete posets are precisely order filters of the connected full heaps of R.M. Green.

Counting Linear Extensions of Posets with Determinants of Hook Lengths

We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entr...

Maximal point spaces of posets with relative lower topology

In domain theory, by a poset model of a T1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ?(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P\?x, x ? P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Max?(P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T1 spaces and order structures. The main results proved in this paper include (i) a T1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T1 space is second-countable if and only if it has an ?-algebraic poset LT-model; (iii) every T1 space has an algebraic dcpo LT-model; (iv) the category of all T1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets.

The symmetric monoidal closed category of cpo M -sets

In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.

Fuzzy uniform domain and its applications

In this paper, firstly, the definition of fuzzy uniform poset is given, and its related properties are discussed. Secondly, the concept of fuzzy uniform complete poset is introduced, and the properties of uniform poset under the condition of complete are explored. Then a new fuzzy uniform way-below relation is introduced, with that the concept of fuzzy uniform domain is given. Finally, the category properties of topological properties of fuzzy uniform domain are discussed.

Unified Characterizations of Minuscule Kac–Moody Representations Built from Colored Posets

R.M. Green described structural properties that "doubly infinite" colored posets should possess so that they can be used to construct representations of most affine Kac–Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets ("full heaps") are analogs of the finite minuscule posets. Here only simply laced Kac–Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficient for such poset-built representations to be produced for smaller subalgebras, especially the "Borel derived" subalgebra. These developments lead to the formulation of unified definitions of finite and infinite colored minuscule and $d$-complete posets. This paper launches a program that seeks to extend the notion of "minuscule representation" to Kac–Moody algebras, and to classify such representations.

Lattices of Irreducibly-derived Closed Sets

This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set ΓSI(X) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of CSI-continuous posets and CSI-prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y, X≅Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.

A uniform approach to completions of posets

For a subset system Z and a subset selection Γ of Z−sets, we introduce two special types of subsets of a poset, called ΔΓ−continuously ⋁−existing subsets and ΔΓ−closed subsets. We define the ZΓ−completion ZΓ(P) to be the set of all ΔΓ−continuously ⋁−existing, ΔΓ−closed subsets of a given poset P. We prove that if Z is subset-hereditary, then (1) the ZΓ−completion ZΓ(P) is the smallest Z−complete subposet containing all principal ideals in the set of all ΔΓ−closed subsets of P; (2) any ΔΓ−continuous function f:P→L mapping into a Z−complete poset L extends uniquely to a Z∨−continuous function from the completion ZΓ(P) to L. The ZΓ−completion includes numerous special cases: the Dedekind-MacNeille completion, the Frink ideal completion, the ideal completion, the Z−completion, the Hoare powerdomain, etc.