Topic
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.
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TL;DR: For appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to is of the power of the continuum.
Abstract: Let be the semigroup variety determined by the identity xm=xm+k. For we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety of the idempotent algebras of the members of form a variety and yields a complete homomorphism of the lattice of subvarieties of onto the lattice of subvarieties of . The lattice contains a ∩-semilattice isomorphic to the ∩-semilattice of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to is of the power of the continuum. For any , ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).
5 citations
01 May 2018
TL;DR: In this article, it was shown that every endomorphism of an infinite-dimensional vector space over a field splits into four idempotents and into four square-zero endomorphisms, a result that is optimal in general.
Abstract: We prove that every endomorphism of an infinite-dimensional vector space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.
5 citations
TL;DR: A rule format for Structural Operational Semantics that guarantees that a unary operator be idempotent modulo bisimilarity and a companion one ensuring that certain terms are idempotsent with respect to some binary operator are presented.
5 citations
TL;DR: -'a is a product of idempotent mappings in T by Theorem 1.1.1, demonstrating that a = 38.
Abstract: that some subset of B is mapped bijectively to a(B). But then f-' a is a mapping in TB. Moreover, ,-6 is noninvertible since a is noninvertible. Hence -'a is a product of idempotent mappings in T by Theorem 1.1. Each of these idempotent mappings can be extended to an idempotent linear mapping from V to V. Let 8 denote the product of these linear mappings. It follows that a(B) = 8(/--1)(B) = 8(B), demonstrating that a = 38. Because both 8 and /3 are products of idempotents, so is a. U
5 citations
TL;DR: It is proved that the class F 2 ( D ) of all idempotent fuzzy subsets of D, where D is a semigroup in which the cancellation laws are valid, forms a complete lattice under the pointwise definition of order.
5 citations