Conservative median algebras and semilattices

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains

On homomorphisms between products of median algebras

International audienceHomorphisms of products of median algebras are studied with particular attention to the case when the codomain is a tree. In particular, we show that all mappings from a product $\mathbf{A}_1\times \cdots \times \mathbf{A}_n$ of median algebras to a median algebra $\mathbf{B}$ are essentially unary whenever the codomain $\mathbf{B}$ is a tree. In view of this result, we also characterize trees as median algebras and semilattices by relaxing the defining conditions of conservative median algebras

Arrow Type Impossibility Theorems over Median Algebras

We characterize trees as median algebras and semilattices by relaxing conservativeness. Moreover, we describe median homomorphisms between products of median algebras and show that Arrow type impossibility theorems for mappings from a product A 1 x...x An of median algebras to a median algebra B are possible if and only if B is a tree, when thought of as an ordered structure

Arrow Type Impossibility Theorems over Median Algebras

We characterize trees as median algebras and semilattices by relaxing conservativeness. Moreover, we describe median homomorphisms between products of median algebras and show that Arrow type impossibility theorems for mappings from a product A 1 x...x An of median algebras to a median algebra B are possible if and only if B is a tree, when thought of as an ordered structure

Median preserving aggregation functions

International audienceA median algebra is a ternary algebra that satisfies every equation satisfied by the median terms of distributive lattices. We present a characterization theorem for ag-gregation functions over conservative median algebras. In doing so, we give a characterization of conservative median algebras by means of forbidden substructures and by providing their representation as chains

On a special class of median algebras

International audienceIn this short note we consider a class of median algebras, called (1, 2 : 3)-semilattices, that is pertaining to cluster analysis. Such median algebras arise from a natural generalization of conservativeness, and their description is given in terms of forbidden substructures

On the number of essential arguments of homomorphisms between products of median algebras

International audienceIn this paper we characterize classes of median-homomorphisms between products of median algebras, that depend on a given number of arguments, by means of necessary and sufficent conditions that rely on the underlying algebraic and on the underlying order structure of median algebras. In particular, we show that a median-homomorphism that take values in a median algebra that does not contain a subalgebra isomorphic to the m-dimensional Boolean algebra as a subalgebra cannot depend on more than m − 1 arguments. In view of this result, we also characterize the latter class of median algebras. We also discuss extensions of our framework on homomorphisms over median algebras to wider classes of algebras

A survey of clones on infinite sets

A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.Comment: 37 page