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: In this paper, it was shown that invertible, linear and idempotent preserving operators on n × n matrices over entire antirings are exactly conjugate actions for n ⩾ 3.
11 citations
Posted Content•
TL;DR: In this paper, an idempotent analogue of the exterior algebra for the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces.
Abstract: We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.
11 citations
Journal Article•
TL;DR: An axiomatic study of certain semirings and related structures that occur in operations research and computer science focuses on the properties A,I,U,G,Z,L and proves that the only implications linking the above properties are essentially those already known.
Abstract: We undertake an axiomatic study of certain semirings and related structures that occur in operations research and computer science. We focus on the properties A,I,U,G,Z,L that have been used in the algebraic study of path problems in graphs and prove that the only implications linking the above properties are essentially those already known. On the other hand we extend those implications to the framework of left and right variants of A,I,U,G,Z,L, and we also prove two embedding theorems. Further generalizations refer mainly to semiring-like algebras with a partially defined addition, which is needed in semantics. As suggested by idempotency (I) and absorption (A), we also examine in some detail the connections between semirings and distributive lattices, which yield new systems of axioms for the latter.
11 citations
TL;DR: In this paper, a pair p and e consisting of a projection p (an idempotent) and an effect e (an element between 0 and 1) in a synaptic algebra was studied.
11 citations
TL;DR: In this paper, the rank and idempotent rank of the semigroup E(X,P) generated by the idempots of a semigroup consisting of all transformations of the finite set $X$ preserving a non-uniform partition was calculated.
Abstract: We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
11 citations