Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

The Relationship Between Two Commutators

Abstract: We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi-affine in such varieties. We refine this by showing that if A is an abelian algebra and (A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.

Some Ordered Sets in Computer Science

Abstract: Strictly speaking the structures to be used are not lattices since as posets they will lack the top (or unit) element, but the adjunction of a top will make them complete lattices. The closure properties as posets, then, are closure underinf of any non-empty subset and sup of directed subsets. A family of subsets of a set closed underintersections of non-empty subfamilies and unions of directed subfamilies is a special type of poset with the closure properties where additionally every element is the directed sup (union) of the finite (“compact”) elements it contains. We call such posets finitary domains. (With a top they are just the well known algebraic lattices.) The continuous domains can be defined as the continuous retracts of finitary domains. A mapping between domains is continuous if it preserves direct sups. A map of a domain into itself is aretraction if it is idempotent. Starting with a finitary domain, the range (= fixed-point set) of a continuous retraction — as a poset — is a continuous domain. Numberless characterizations of continuous domains, both topological and order-theoretic, can be found in [2]. For the most part in the lectures we shall concentrate on the finitary domains, but the continuous domains find an interest as a generalization of interval analysis and by the connection with spaces of upper-semicontinuous functions.

Combinatorics of the free Baxter algebra

Abstract: We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.