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It is known that the so-called Beck's conjecture, i.e. the equality of the finite clique and chromatic numbers of a zero-divisor graph, holds for partially ordered sets Halaš and Jukl (Discrete Math 309(13):4584–4589, 2009). In this paper we present a simple direct proof of this fact. Also, the case when the finiteness assumption of the clique number is omitted is investigated. We have shown that the conjecture fails in general and a bunch of counterexamples is presented. [ABSTRACT FROM AUTHOR]

Given a semigroup S , for each Green's relation K ∈ { L , R , J , H } on S , the K -height of S , denoted by H K (S) , is the height of the poset of K -classes of S. More precisely, if there is a finite bound on the sizes of chains of K -classes of S , then H K (S) is defined as the maximum size of such a chain; otherwise, we say that S has infinite K -height. We discuss the relationships between these four K -heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite L -height if and only if it has finite R -height if and only if it has finite J -height. In fact, for a stable semigroup S , if H L (S) = n then H R (S) ≤ 2 n − 1 and H J (S) ≤ 2 n − 1 , and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2 ≤ H L (S) < ∞ and 2 ≤ H R (S) < ∞ , then H J (S) ≤ H L (S) + H R (S) − 2. We also show that for each n ∈ N there exists a semigroup S such that H L (S) = H R (S) = 2 n + n − 3 and H J (S) = 2 n + 1 − 4. By way of contrast, we prove that for a regular semigroup the L -, R - and H -heights coincide with each other, and are greater or equal to the J -height. Moreover, in a stable, regular semigroup the L -, R -, H - and J -heights are all equal. [ABSTRACT FROM AUTHOR]

We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group (Z 2 4 × S 4) ⋊ α Z 2 . Though unable to prove it, we solidify this conjecture by showing that the automorphism group behaves as expected by the conjecture on the first few levels of the poset in question. With the use of computer calculation we analyze the first four levels holding 3160 directed graphs. [ABSTRACT FROM AUTHOR]

In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if A T = B for a nonnegative integer matrix T , then we say A ≤ B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k -Catalan arrays, k -Schröder arrays, and k -Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur. [ABSTRACT FROM AUTHOR]

In this note we exhibit various ways of obtaining a wide range of soluble partially ordered sets. These are the posets on which every transitive system can be completed. In particular, algebraic operations such as the sums and products of posets are proven to be powerful tools to create new soluble posets. [ABSTRACT FROM AUTHOR]

We introduce the notion of topogenous orders on a poset X to be certain endomaps on X. We build on a Galois connection between endomaps and binary relations on X and study relationships between endomap properties and corresponding relational properties. In particular, we determine the topogenous orders that are in a one-to-one correspondence with (idempotent) closure operators. Extending our considerations to the categorical level, we find a cartesian closed category of topogenous systems. [ABSTRACT FROM AUTHOR]

Based on the classical works of Clifford inducing partial order from semigroups, recently, Gupta and Jayaram explored the order ⊑F from an associative operation F through local left identity (LLI). Inspired by their works, we further present an order ⊑∗ F obtained from non-commutative operation F which has the local right identity (LRI) since the non-commutativity of F implies that the local left and right identity may be different for each element, which means that both orders may not coincide in the same domain. Firstly, we determine an equivalent characterization for two orders induced by non-commutative operation F. Secondly, we investigate both orders induced by semi-t-operators and deeply study their properties. Finally, we characterize both orders obtained from semi-uninorm (resp. semi-nullnorm) under the condition that semi-uninorm (resp. semi-nullnorm) is locally continuous. [ABSTRACT FROM AUTHOR]

We introduce a notion of proper morphism for schematic finite spaces and prove the analog of Grothendieck’s finiteness theorem for it. The techniques we employ, which further develop the theory of schematic spaces and proschemes, are ultimately founded on descent properties of flat epimorphisms of rings that are applicable in other situations in order to weaken finite presentation requirements. [ABSTRACT FROM AUTHOR]

To any poset P, we associate a convex cone called a braid cone. We also associate a fan and study the toric varieties the cone and fan define. The fan always defines a smooth toric variety X P , while the toric variety U P of the cone may be singular. We show that X P ⤏ U P is a crepant resolution of singularities if and only if P is bounded. Next, we aim to determine when U P is Q -Gorenstein or Gorenstein. We prove that whether or not U P is Q -Gorenstein or Gorenstein depends only on the biconnected components of the Hasse diagram of P. In the case that P has a minimum or maximum element, we show that the Gorenstein property of U P is completely determined by the Möbius function of P. We also provide a recursive method that determines if U P is Q -Gorenstein or Gorenstein in this case. We conjecture that U P is Gorenstein if and only if it is Q -Gorenstein. We verify this conjecture for posets of length 1 and also for posets with a minimum or maximum element. [ABSTRACT FROM AUTHOR]

In this study, we compute some examples of sectional category secat(f) and sectional number sec(f) for continuous maps f related to finite spaces. Moreover, we introduce an invariant secatk(f) for a map f between finite spaces using the k-th barycentric subdivision and show the equality secatk(f) = secat(B(f)) for sufficiently large k, where B(f) is the induced map on the associated polyhedra. [ABSTRACT FROM AUTHOR]

The Vlsekriterijumska Optimizacija I Komprosmisno Resenie (VIKOR) method to some extent modifies the utility function to a value function that can consider different risk preferences. However, the weight and risk attitude parameters involved in the model are difficult to determine, which limits its application. To overcome this problem, a Poset-VIKOR model is proposed. A partial order set is a non-parametric decision-making method. Through the combination of partial order set and VIKOR model, the parameters can be “eliminated”, and a robust method that can run the model is obtained. This method uses the Hasse diagram to express the evaluation results, which can not only directly display the hierarchical and clustering information, but also show the robustness characteristics of the alternative comparison. [ABSTRACT FROM AUTHOR]

We show that height h posets that have planar cover graphs have dimension O (h 6). Previously, the best upper bound was 2 O (h 3). Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes K 5 as a minor. [ABSTRACT FROM AUTHOR]

The so-called matrix properties of finitely complete categories are a special type of exactness properties that can be encoded as extended matrices of integers. The relation of entailment of matrix properties gives an interesting preorder on the set of such matrices, which can be investigated independently of the category-theoretic considerations. In this paper, we conduct such investigation and obtain several results dealing with binary matrices, i.e., when the only integer entries in the matrix are 0 or 1. Central to these results is a complete description of the preorder in the case of a special type of binary matrices, which we call diagonal matrices, and which include matrices that define Mal'tsev, majority and arithmetical categories. [ABSTRACT FROM AUTHOR]

Abstract To achieve an objective and reasonable judgment on slope stability, the partially ordered set (Poset) theory is applied to the slope stability evaluation against the engineering background of an open pit slope. From the four aspects of rock mass structural characteristics, slope morphology, environment conditions and engineering conditions, 24 typical evaluation indices such as cohesion, internal friction angle, elastic modulus and Poisson’s ratio are selected. Based on the Poset theory, a Poset evaluation model is established. According to the slope stability evaluation indices and the risk grade classification criteria, the slope stability risk grade is divided into four grades: stable (Grade I), generally stable (Grade II), slightly stable (Grade III) and unstable (Grade IV). The index data are normalized and the weight ranking information is used. Implicit weighting is carried out by means of accumulation. The comparison relation matrix, the Hasse matrix and the Hasse diagram are constructed for each factor layer and comprehensive index layer, respectively. This method was utilized to evaluate the risk grade of an engineering example, determine the slope risk grade, and compare with the actual situation. The results show that the slope to be evaluated is generally stable (Grade II) under the comprehensive evaluation indices of various factors, and the assessed stability condition is consistent with the actual situation. These verify the rationality and applicability of the proposed method and provide a new insight for accurately identifying the stability condition of open-pit slopes.

For each poset P, we construct a polytope A (P) called the P-associahedron. Similarly to the case of graph associahedra, the faces of A (P) correspond to certain nested collections of subsets of P. The Stasheff associahedron is a compactification of the configuration space of n points on a line, and we recover A (P) as an analogous compactification of the space of order-preserving maps P → R . Motivated by the study of totally nonnegative critical varieties in the Grassmannian, we introduce affine poset cyclohedra and realize these polytopes as compactifications of configuration spaces of n points on a circle. For particular choices of (affine) posets, we obtain associahedra, cyclohedra, permutohedra, and type B permutohedra as special cases. [ABSTRACT FROM AUTHOR]

It is already known that the 1-Poset and 2-Poset Cover Problems are in 𝑷. In this paper, we extended the previous results and devised an algorithm for the 𝒌-Poset Cover Problem, for any 𝒌 number of posets that cover the input. The algorithm runs in 𝑶(𝒎2k 𝒏2), where 𝒎 and 𝒏 are the input size. With this running time, we can say that the problem belongs to XP (slicewise polynomial). The algorithm runs efficiently for small fixed 𝒌 but runs exponentially for large 𝒌. While the algorithm running time has yet not to be efficient for large 𝒌, we have shown a significant improvement from a brute-force solution, which runs in 2𝒎 𝒏2 + 𝑶(􀷌𝒌 𝒊 􀰸 1 2𝒎𝒊𝒏) - exponential even for small fixed [ABSTRACT FROM AUTHOR]

The participation of women in digital society is essential to achieving the objectives of Agenda 2030 and an essential component in the EU strategy for the digital transition. This article applies a poset-based approach to the European Women in Digital (WiD) Scoreboard, to examine the digital inclusion of women in EU Member States and in the UK. The poset methodology enables us to identify the most significant indicators for each of the dimensions of the Scoreboard, considering the EU-28 as well as different clusters of countries, and to construct a new ranking that deals with the shortcomings of the aggregative approaches, the pre-treatment of data and the full compensability effect generated by arithmetic means. Our results show that two indicators, STEM graduates and the unadjusted pay gap, are the most significant for achieving the digital inclusion of women. Our research contributes to better understanding the dynamics and the factors facilitating the digital inclusion of women in the EU-28 Member States, providing a clustering of EU countries into four performance groups depending on the digital inclusion of women. It also contributes to the design of more targeted and effective policies for integrating gender equality into the EU digital transition strategy. [ABSTRACT FROM AUTHOR]

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $\delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $\delta$-cliff. We provide necessary and sufficient conditions on $\delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $\delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.Mathematics Subject Classifications: 05E99, 06A07, 06A11, 16T30

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to study in which posets some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again.

A topological duality result is established for the category of distributive c-posets defined in this paper, as well as for some of its important full subcategories. All duality results presented extend the well-known topological duality result obtained by M. H. Stone for the category of distributive (0, 1)-lattices. [ABSTRACT FROM AUTHOR]

The breadth of a Lie algebra L is defined to be the maximal dimension of the image of a d x = [ x , − ] : L → L , for x ∈ L. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide combinatorial breadth formulas for members of each family. [ABSTRACT FROM AUTHOR]

Let (P , ≤) be an atomic poset with the least element 0. The complement of the intersection graph of ideals of P , denoted by Γ (P) , is defined to be a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J = { 0 }. In this paper, we consider the complement of the intersection graph of ideals of a poset. We prove that Γ (P) is totally disconnected or diam (Γ (P) \) ∈ { 1 , 2 , 3 } , where is the set of all isolated vertices of Γ (P). We show that g r (Γ (P)) ∈ { 3 , 4 , ∞ }. Also, we characterize all posets whose complement of the intersection graph is forest, unicyclic or complete r -partite graph. Among other results, we prove that Γ (P) is weakly perfect; and it is perfect if and only if | Atom (P) | ≤ 4. Finally, we show that Γ (P) is class 1 , where P = Atom (P) ∪ { 0 }. [ABSTRACT FROM AUTHOR]

In this study, a new method, called g-sum, is presented to join two posets together that generalizes the linear sum of two posets. Idempotent uninorms on a bounded chain are in one-to-one correspondence with special linear orders on it and g-sum can be used to construct such special linear orders. [ABSTRACT FROM AUTHOR]

The identification of territorial clusters where the population suffers from worse health conditions is an important topic in social epidemiology, in order to identify health inequalities in cities and provide health policy interventions. This objective is particularly challenging because of the mechanism of self-selection of individuals into neighborhoods, which causes selection bias. The aim of this paper consists in the identification of neighborhood clusters where elderly people living in Turin, a city in north-western Italy, are exposed to an increased risk of hospitalized fractures. The study is based on administrative data and is a retrospective, observational cohort study. It is composed by a first phase, in which the individual confounding variables are balanced across neighborhoods in order to make them comparable, and a second phase in which the neighborhoods are aggregated into clusters characterized by significantly higher health risk. In the first phase we exploited a balancing technique based on partially ordered sets (poset), called Matching on poset based Average Rank for Multiple Treatments (MARMoT). On the balanced dataset, we used a spatial scan to identify the presence of clusters and we checked whether the risk of fracture is significantly higher in some contiguous areas. The combination of both MARMoT procedure and spatial scan makes it possible to highlight two clusters of neighborhoods in Turin where the risk of incurring hospitalized fractures for elderly people is significantly higher than the mean. These results could have important implications for the implementation of health policies. [ABSTRACT FROM AUTHOR]

In his 2008 thesis [16] , Tateno claimed a counterexample to the Bonato–Tardif conjecture regarding the number of equimorphy classes of trees. In this paper we revisit Tateno's unpublished ideas to provide a rigorous exposition, constructing locally finite trees having an arbitrary finite number of equimorphy classes; an adaptation provides partial orders with a similar conclusion. At the same time these examples also disprove conjectures by Thomassé and Tyomkyn. [ABSTRACT FROM AUTHOR]

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a bounded poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra. [ABSTRACT FROM AUTHOR]

In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the 2-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line Re (z) = − 1 2 . A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope Q and its dual Q ∨ . We discuss the CL-ness and realness of Q and Q ∨ in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation. [ABSTRACT FROM AUTHOR]

Combinatorial properties of finite posets connected with the positivity of their quadratic Tits form (playing an important role in the theory of matrix representations of posets) are investigated. We determine the coefficients of transitivity for all minimal posets with nonpositive quadratic Tits form [these posets are called P -critical and their number is equal to 75 (up to isomorphism and duality)]. Some relationships between these coefficients and the heights of posets are established. [ABSTRACT FROM AUTHOR]

The concept of a tolerance relation, shortly called tolerance, was studied on various algebras since the seventies of the twentieth century by B. Zelinka and the first author (see e.g. [6] and the monograph [1] and the references therein). Since tolerances need not be transitive, their blocks may overlap and hence in general the set of all blocks of a tolerance cannot be converted into a quotient algebra in the same way as in the case of congruences. However, G. Czédli ([7]) showed that lattices can be factorized by means of tolerances in a natural way, and J. Grygiel and S. Radeleczki ([8]) proved some variant of an Isomorphism Theorem for tolerances on lattices. The aim of the present paper is to extend the concept of a tolerance on a lattice to posets in such a way that results similar to those obtained for tolerances on lattices can be derived. [ABSTRACT FROM AUTHOR]

The bivariate chromatic polynomial χG(x, y) of a graph G=(V, E), introduced by Dohmen-Pönitz-Tittmann (2003), counts all x-colorings of G such that adjacent vertices get different colors if they are ≤y. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses χG(x, y) as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for χG(x, y). [ABSTRACT FROM AUTHOR]

Since its introduction, researching malware has had two main goals. On the one hand, malware writers have been focused on developing software that can cause more damage to a targeted host for as long as possible. On the other hand, malware analysts have as one of their main purposes the development of tools such as malware detection systems (MDS) or network intrusion detection systems (NIDS) to prevent and detect possible threats to the informatic systems. Obfuscation techniques, such as the encryption of the virus's code lines, have been developed to avoid their detection. In contrast, shallow machine learning and deep learning algorithms have recently been introduced to detect them. This paper is devoted to some theoretical implications derived from these investigations. We prove that hidden algebraic structures as equipped posets and their categories of representations are behind the research of some infections. Properties of these categories are given to provide a better understanding of different infection techniques. [ABSTRACT FROM AUTHOR]

Associated with any composition β = (a , b , ...) is a corresponding fence poset F (β) whose covering relations are x 1 ⊲ x 2 ⊲ ... ⊲ x a + 1 ⊳ x a + 2 ⊳ ... ⊳ x a + b + 1 ⊲ x a + b + 2 ⊲ .... The distributive lattice L (β) of all lower order ideals of F (β) is important in the theory of cluster algebras. In addition, its rank generating function r (q ; β) is used to define q-analogues of rational numbers. Kantarcı Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth, and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that r (q ; β) is unimodal. We show that, when β has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F (β) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Kantarcı Oğuz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result, as well. We end with some questions and conjectures raised by this work. [ABSTRACT FROM AUTHOR]