The generation of clones with majority operations

A theorem by Baker and Pixley implies that any clone on a finite set is finitely generated if it contains a near-unanimity operation. This raises the question of what arity the generating operations must have. In this paper, we solve the last open bits of this problem for the majority case by showing that 5 and 8 are the smallest integers k such that every clone with a majority operation on a 3 and 4-element set, respectively, is generated by its k-ary part.     

On clones generated by primitive-positive operations of Tarski's relation algebras

Received November 29, 1996; accepted in final form September 5, 1997.     

On the (un)decidability of a near-unanimity term

We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem. While working on this paper, the author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant nos. T 37877 and K 60148.     

Minimal clones generated by majority operations

The minimal majority functions of the four-element set are determined. Received October 10, 1997; accepted in final form February 8, 2000.     

Equivalence of operations with respect to discriminator clones

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.     

Properties of the quasilinear clones containing creative functions

We study the problem of characterizing clones on a three-element set by hyperidentities. We prove that there exists a hyperidentity separating any clone of quasilinear functions defined on the set {0, 1, 2} each of them is either a selector or such that all its values belong to {0, 1} from any noncreative clone constituted by such functions incomparable with the initial clone.     

On the logical equivalence of functional clones

We study the equivalence relations between functional clones which are based on the coincidence of their quantifier-free formal subsets of the corresponding universal algebras.     

On the structure and representation of clones

Presented by Walter Taylor.     

On the cooperation of recycling operations

This paper addresses two important issues that may affect the operations efficiency in the recycling industry. First, the industry contains many small-scale and inefficient recycling firms, especially in developing countries. Second, the output from recycling a waste product often yields multiple recycled products that cannot all be sold efficiently by a single firm. To address these two issues, this paper examines how different firms can cooperate in their recycling and pricing decisions using cooperative game theory. Recycling operations under both joint and individual productions with different cost structures are considered. Decisions include the quantity of waste product to recycle and the price at which to sell each recycled product on each firm’s market. These decisions can be made jointly by multiple cooperating firms to maximize total profit. We design allocation schemes for maximized total profit to encourage cooperation among all firms. Managerial insights are provided from both environmental and economic perspectives.     

On the semigroup of closure operations

Communicated by J.S. Ponizovskii     

On essential variables of dual operations and the consequences for operations

We investigate essential variables of dual operations (or co-operations) in a rather general category-theoretic setting. During this investigation, we collect some results about the essential arities and arity gaps of dual operations in concrete categories. Afterwards, we use duality theory to show that this approach and the obtained results can also be beneficial for questions regarding essential variables of operations on sets.     

The existence of a near-unanimity function is decidable

We prove that the following problem is decidable: given a finite set of relations on a finite set, decide whether this set admits a near-unanimity function. The proof is based on the upper bound on the minimal arity of a near-unanimity function admitted by a set of relations. Also, we give examples that show that the upper bound cannot be significantly improved.     

On Boolean primitive positive clones

Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25 Boolean primitive positive clones in a notoriously unavailable article. This paper presents a new proof of his result, relating it to Post's lattice and exhibiting finite bases for those clones.     

On algebraic properties of monotone clones

Let R ₈ denote the 8-element bounded tower. G. Tardos has shown that C(R ₈), the clone of all monotone functions on R ₈, is not finitely generated. In this paper we show that the clone of all nonsurjective functions is finitely generated.     

On monotone hull operations

We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ‐ideals on the plane. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the local closure of clones on countable sets

We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function \({f : \mathbb{N} \rightarrow \mathbb{N}}\) such that the n-ary part of C is equal to the n-ary part of Pol \({{\rm Inv}^{[f(n)]} C}\), where \({{\rm Inv}^{[f(n)]} C}\) denotes the set of f(n)-ary invariant relations of C.