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.

Defects in Kitaev models and bicomodule algebras

Abstract: We construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. More specifically, we show that one can consider cell decompositions of surfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeled by semisimple bicomodule algebras. We introduce an algebra whose representations label the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in the absence of defects. In this way we generalize the algebraic structure underlying the standard Kitaev model without defects or boundaries, where all 1-cells and 2-cells are labeled by a single Hopf algebra and where point defects are labeled by representations of its Drinfeld double. In the standard case, commuting local projectors are constructed using the Haar integral for semisimple Hopf algebras. A central insight we gain in this paper is that in the presence of defects and boundaries, the suitable generalization of the Haar integral is given by the unique symmetric separability idempotent for a semisimple (bi-)comodule algebra.

One-generated semirings and additive divisibility

Abstract: We study the structure of one-generated semirings from the symbolical point of view and their connections to numerical semigroups. We prove that such a semiring is additively divisible if and only if it is additively idempotent. We also show that every at most countable commutative semigroup is contained in the additive part of some one-generated semiring.

Local structure of idempotent algebras II

Abstract: We refine and advance the study of the local structure of idempotent finite algebras started in [ABulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004] We introduce a graph-like structure on an arbitrary finite idempotent algebra including those admitting type 1 We show that this graph is connected, its edges can be classified into 4 types corresponding to the local behavior (set, semilattice, majority, or affine) of certain term operations We also show that if the variety generated by the algebra omits type 1, then the structure of the algebra can be `improved' without introducing type 1 by choosing an appropriate reduct of the original algebra Taylor minimal idempotent algebras introduced recently is a special case of such reducts Then we refine this structure demonstrating that the edges of the graph of an algebra omitting type 1 can be made `thin', that is, there are term operations that behave very similar to semilattice, majority, or affine operations on 2-element subsets of the algebra Finally, we prove certain connectivity properties of the refined structures This research is motivated by the study of the Constraint Satisfaction Problem, although the problem itself does not really show up in this paper