Kira Adaricheva

• PhD
• Professor at Hofstra University

Improving undergraduate success in STEM requires identifying actionable factors that impact student outcomes, allowing institutions to prioritize key leverage points for change. We examined academic, demographic, and institutional factors that might be associated with graduation rates at two four-year colleges in the northeastern United States usin...

Implicational bases (IBs) are a well-known representation of closure spaces and their closure lattices. This representation is not unique, though, and a closure space usually admits multiple IBs. Among these, the canonical base, the canonical direct base as well as the $D$-base aroused significant attention due to their structural and algorithmic p...

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat [1] and the Polymath REU (2020), continues to investigate representations of convex geometries with small convex dimension by convex shapes on the plane and in spaces of higher dimension. In particular, we answer in th...

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (Discrete Math 342(N3):726–746, 2019) and the Polymath REU 2020 team (Convex geometries representable by at most 5 circles on the plane. arXiv:2008.13077 ), continues to investigate representations of convex geometries...

The sea breeze is a phenomenon frequently impacting Long Island, New York, especially during the spring and early summer, when land surface temperatures can exceed ocean temperatures considerably. The sea breeze influences daily weather conditions by causing a shift in wind direction and speed, limiting the maximum temperature, and occasionally ser...

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat (2019) and the Polymath REU (2020), continues to investigate representations of convex geometries with small convex dimension by convex shapes on the plane and in spaces of higher dimension. In particular, we answer in...

The dominating graph of a graph H has as its vertices all dominating sets of H, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of H. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to...

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of K. Adaricheva and M. Bolat (2016) and the Polymath REU 2020 team, continues to investigate representations of convex geometries on a 5-element base set. It introduces several properties: the opposite property, nested triangle property, are...

There is a natural closure operator Γ on the lattice \(\text{L}_{\text{q}}(\mathcal K)\) of subquasivarieties of a quasivariety \(\mathcal K\): for \(\mathcal Q \leq \mathcal K\) let \(\varGamma (\mathcal Q) = \mathcal K \cap \mathbb {HSP}(\mathcal Q) = \mathcal K \cap \mathbb {H}(\mathcal Q)\). The map Γ is called the natural equaclosure operator...

In this chapter, we turn to the case of representing a finite lattice by subsemilattices with operators when the join irreducibles are not least in their γ-classes. There is no algorithm for this case, but there are reasonably systematic and effective ad hoc methods.

In this chapter, we concentrate on a method for representing pairs (L, γ), where L is a finite lower bounded lattice and γ an equaclosure operator on it, as \((\text{L}_{\text{q}}(\mathcal K),\varGamma )\) for some quasivariety \(\mathcal K\) and its natural equaclosure operator Γ.

A finite lattice may be regarded as a join semilattice with [Formula: see text]. Using this viewpoint, we give algorithms for testing semidistributivity, provide a new characterization of convex geometries, and characterize congruence lattices of finite join semidistributive lattices.

A preclop is a closure operator on a lattice that satisfies some of the known properties of the equational closure operator on subquasivariety lattices. In this chapter we look at preclops on finite lattices and give an algorithm to decide if a finite lattice supports a preclop.

In this chapter and the next, we address the question: Given a pair (L, γ) consisting of a finite lower bounded lattice and a preclop, when can we find a finite semilattice S and a set H of operators such that \(\mathbf L \cong \operatorname {Sub}(\mathbf S,\wedge ,1,H)\) with γ corresponding to the natural equaclosure operator Γ? There is an algor...

In this chapter we consider conditions under which, given a finite join semidistributive lattice, the linear sum \(\mathbf 1 + \mathbf L \cong \operatorname {L}_{\text{q}}(\mathcal K)\) for some quasivariety \(\mathcal K\) of structures in a language with equality. In particular, if L is isomorphic to the lattice of H-closed subsets of an algebraic...

In this chapter we will develop the fundamental notions of varieties and quasivarieties, starting near the beginning (just past set theory). The key idea is to link logical theories (equational theories or implicational theories) with models (varieties or quasivarieties). The tools are well-established: homomorphisms, subalgebras, direct products,...

We show that every distributive dually algebraic lattice can be represented as Sp(S, H) with S an algebraic lattice and H a monoid of operators. As a consequence, every linear sum 1 + D with D distributive and dually algebraic is isomorphic to a lattice of subquasivarieties \(\text{L}_{\text{q}}(\mathcal K)\) with equality. Moreover, every distribu...

In this chapter we propose a number of open problems about subquasivariety lattices.

The dominating graph of a graph $G$ has as its vertices all dominating sets of $G$, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of $G$. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also prove that the dominating graph of a cycle...

This study applied two mathematical algorithms, lattice up-stream targeting (LUST) and D-basis, to the identification of prognostic signatures from cancer gene expression data. The LUST algorithm looks for metagenes, which are sets of genes that are either overexpressed or underexpressed in the same patients. Whereas LUST runs unsupervised by clini...

The dominating graph of a graph G has as its vertices all dominating sets of G, with an edge between two dominating sets if one can be obtained from the other by adding or deleting a single vertex of G. This is an example of a reconfiguration graph. This paper gives a brief introduction to the study of reconfiguration of dominating sets, and to the...

A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we ma...

Convex geometry is a closure space \((G,\phi )\) with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necess...

We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the D-basis introduced in Adaricheva, Nation, Rand [4]. Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, when the closure syste...

We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the $D$-basis introduced in Adaricheva, Nation, Rand [Disc.Appl.Math. 2013]. Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, w...

We develop a new approach for distributed computing of the association rules of high confidence on the attributes/columns of a binary table. It is derived from the D-basis algorithm developed by K.Adaricheva and J.B.Nation (Theoretical Computer Science, 2017), which runs multiple times on sub-tables of a given binary table, obtained by removing one...

Convex geometry is a closure space $(G,\phi)$ with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary...

We develop a new approach for distributed computing of the association rules of high confidence in a binary table. It is derived from the D-basis algorithm in K. Adaricheva and J.B. Nation (TCS 2017), which is performed on multiple sub-tables of a table given by removing several rows at a time. The set of rules is then aggregated using the same app...

A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1-element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many min...

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice $\Omega(\eta)$, that does not appear among minimal obstructions to...

Sets of implications defining closure systems are used as a standard way to represent knowledge, and the search of implicational systems satisfying some criteria constitutes one of the most active topics in the study of closure systems and their applications. Here, we focus on the generation of the D-basis, known to be an ordered direct basis, allo...

Around 1960, Bjarni Jónsson was analyzing the structural consequences of P. M. Whitman’s solution to the word problem for free lattices [448], and made the following observation. Whitman had shown that every element in Free(X) can be represented by a unique (up to associativity and commutativity) shortest term ω.

The origin of convex geometries lies in combinatorics, and the goal of the study of finite convex geometries was to develop the combinatorial abstraction of convexity. Similarly, the theory of matroids is a combinatorial abstraction of independent sets; see the survey of B. Dietrich [125]. Since both abstractions can be formulated in the framework...

In Section 4-2.2 we discussed the fundamental connection between closure systems and sets of implications. In this chapter, we will look into the canonical forms of representations of a closure system by implications. Most of the results are inspired by the structure of the closure lattice and its properties. In particular, we will be concerned wit...

In this section we consider one of the central constructions producing joinsemidistributive lattices, which turned out to be crucial in connecting many aspects of these lattices. The goal of this section is to present Sp(PowX), the lattice of algebraic subsets of the powerset lattice of some set X. We will need several definitions and lemmas on the...

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara, M.Nakamura and Y.Okamoto (2005). Allowing circles rather than points, as was suggested...

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara, M.Nakamura and Y.Okamoto (2005). Allowing circles rather than points, as was suggested...

The ALEKS computer system, well-known on the American continent as an individualized teaching tool and placement test into a series of high school and college level courses, was implemented by the Mathematics Department of Nazarbayev University, most likely as the first experiment of this sort in the heart of Asia. This paper summarizes the results...

George Grätzer’s Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would...

We introduce the parameter of relevance of an attribute of a binary table to another attribute of the same table, computed with respect to an implicational basis of a closure system associated with the table. This enables a ranking of all attributes, by relevance parameter to the same fixed attribute, and, as a consequence, reveals the implications...

Discovery of (strong) association rules, or implications, is an important task in data management, and it finds application in artificial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a specific set of implications, called the D-basis, that provides a representation for a reduced binary table, ba...

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.

Representation of convex geometry as an appropriate join of compatible total orderings of the base set can be achieved, when closure operator of convex geometry is algebraic, or finitary. This bears to the finite case proved by P.H.~Edelman and R.E.~Jamison to the greater extent than was thought earlier.

Modes are idempotent and entropic algebras. While the mode structure of sets of submodes has received considerable attention in the past, this paper is devoted to the study of mode structure on sets of mode homomorphisms. Connections between the two constructions are established. A detailed analysis is given for the algebra of homomorphisms from su...

Let L be a join-distributive lattice with length n and width(Ji L) \leq k. There are two ways to describe L by k-1 permutations acting on an n-element set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a recent lattice theoretical way of the second author. We prove that these two approaches are equivalent. Also, we characte...

Convex geometries form a subclass of closure systems with unique criticals, or UC-systems. We show that the F-basis introduced in [1] for UC-systems, becomes optimum in convex geometries, in two essential parts of the basis. The last part of the basis can be optimized, when the convex geometry either satisfies the Carousel property, or does not hav...

We show that every optimum basis of a finite closure system, in D.Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce a K-basis of a general closure system, which is a refinement of the canonical...

Closure system on a finite set is a unifying concept in logic programming, relational data bases and knowledge systems. It can also be presented in the terms of finite lattices, and the tools of economic description of a finite lattice have long existed in lattice theory. We present this approach by describing the so-called D-basis and introducing...

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new restrictions on the natural quasi-interior oper...

Part I proved that for every quasivariety K of structures (which may have both operations and relations) there is a semilattice S with operators such that he lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). It is known that if S is a join semilattice with 0 (and no operato...

We call a complete lattice perfect if it is a sublattice of a lattice of the form Sp(A), where A is an algebraic lattice and Sp(A) stands for the lattice of algebraic subsets of A. The problem of the description of perfect lattices is motivated by the fact that lattices of subquasivarieties are perfect. In our paper, we describe a new class of per...

An assosiahedron $\mathcal{K}^n$, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in $\mathbf{R}^{n}$, forming $(n-1)$-dimensional polytope. A permutahedron $\mathcal{P}^n$ is a polytope of dimension $(n-1)$ in $\mathbf{R}^{n}$ with vertices forming va...

A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in $n$-dimensional vector space and their finite sub-geometries satisfy the $n$-Carou...

Belief change studies how to update knowledge bases used for reasoning. Traditionally belief revision has been based on full propositional logic. However, reasoning with full propo-sitional knowledge bases is computationally hard, whereas reasoning with Horn knowledge bases is fast. In the past several years, there has been considerable work in bel...

The Edelman–Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman–Jamison problem is equivalent to the well known NP-hard order type problem. The relation to the realizability of oriented matroids is clarified.

Let V be a variety of algebras. We specify a condition (the so-called generalized entropic property), which is equivalent to the fact that for every algebra A ∈ V, the set of all subalgebras of A is a subuniverse of the complex algebra of the subalgebras of A. The relationship between the generalized entropic property and the entropic law is invest...

The Edelman-Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman-Jamison problem is equivalent to the well known NP-hard order type problem.

The least element 0 of a finite meet semi-distributive lattice is a meet of meet-prime elements. We investigate conditions under which the least element of an algebraic, meet semi-distributive lattice is a (complete) meet of meet-prime elements. For example, this is true if the lattice has only countably many compact elements, or if |L| < 2ℵ0, or i...

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, – the class of all finite lattices; – the class of all finite lower bounded lattices (solved by the first author's earlier work); – the class o...

Lattices in the variety LB(k)\mathcal{L}\mathcal{B}(k) of lower bounded lattices of rank k are characterized. A sufficient condition for a lattice to be lower bounded is given, and used to produce a new example of a non-finitely-generated lower bounded lattice. Lattices that are subdirect products of finite lower bounded lattices are characterized...

This survey article tackles different aspects of lattices of algebraic subsets, with the emphasis on the following: the theory of quasivarieties, general lattice theory and the theory of closure spaces with the anti-exchange axiom.

We give two sufficient conditions for the lattice Co(R^n,X) of relatively convex sets of n-dimensional real space R^n to be join-semidistributive, where X is a finite union of segments. We also prove that every finite lower bounded lattice can be embedded into Co(R^n,X), for a suitable finite subset X of R^n.

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also...

We study the structure of algebraic -closed subsets of an algebraic lattice L, where is some Browerian binary relation on L, in the special case when the lattice of such subsets is an atomistic lattice. This gives an approach to investigate the atomistic lattices of congruence-Noetherian quasivarieties.

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded...

It is shown that the several properties defining the class of finite lower bounded lattices are not equivalent in general. We establish also a new structural property of the class of lower bounded lattices.

We consider noncomplete continuous and algebraic lattices and prove that finitely generated free lattices are algebraic. We also study the Lawson topology, the second most important topology in the theory of continuous domains, on finitely presented lattices. In particular, we prove that algebraic finitely presented lattices are linked bicontinuous...

Hypertension can be classified as either Mendelian hypertension or essential hypertension, on the basis of the mode of inheritance. The Mendelian forms of hypertension develop as a result of a single gene defect, and as such are inherited in a simple Mendelian manner. In contrast, essential hypertension occurs as a consequence of a complex interpla...

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congru...

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as Lq(K) for some algebraic quasivariety K; (2) L is represented as SΛ (A) for some algebraic lattice A which sa...

The purpose of this note is to present the following two results that were announced in [2].

G. Zhitomirskii gave a description of congruence lattices of semilattices in the second-order language. Here we describe finite lattices belonging to this class in terms of properties shared by coatoms (minimal nonidentity elements of a lattice). As a consequence, a characterization of finite semilattices with isomorphic congruence lattices is obta...

We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of an earlier paper by the third author and V. I. Tumanov [Algebra Logika 19, 59-80 (1980; Zbl 0472.08011)].