Related papers: A survey of clones on infinite sets
The classical XY model is a lattice model of statistical mechanics notable for its universality in the rich hierarchy of the optical, laser and condensed matter systems. We show how to build complex structures for machine learning based on…
We review a few results concerning interpolation of monotone functions on infinite lattices, emphasizing the role of set-theoretic considerations. We also discuss a few open problems.
This paper studies the configuration space of all possible positions of a linkage in R^n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number…
A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen…
We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations…
Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.)…
The set of all transformation monoids on a fixed set of infinite cardinality \lambda, equipped with the order of inclusion, forms a complete algebraic lattice Mon(\lambda) with 2^{\lambda} compact elements. We show that this lattice is…
Programmers often reuse code from source code repositories to reduce the development effort. Code clones are candidates for reuse in exploratory or rapid development, as they represent often repeated functionality in software systems. To…
Equations over linearly ordered semilattices are studied. For any equation $t(X)=s(X)$ we find irreducible components of its solution set and compute the average number of irreducible components of all equations in $n$ variables.
Traces and their extension called combined traces (comtraces) are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are…
Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for…
There are a great many clone detection tools proposed in the literature. In this paper, we investigate the state of clone detection tool evaluation. We begin by surveying the clone detection benchmarks, and performing a multi-faceted…
Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely…
A genoid is a category of two objects such that one is the product of itself with the other. A genoid may be viewed as an abstract substitution algebra. It is a remarkable fact that such a simple concept can be applied to present a unified…
We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm…
We study connections between closure operators on an algebra $(A,\Om)$ and congruences on the extended power algebra defined on the same algebra. We use these connections to give an alternative description of the lattice of all subvarieties…
The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\mathbf{2} := \{0,1\}$, the set $\mathcal{I}(C)$ of all partial clones on $\mathbf{2}$ whose total component is $C$, is either finite or…
We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic…
Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponing complete lattice of sets. A function mappping into the preordered set is extended to a…