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: It is proved that in Sh(Q), a finitely generated projective module is free (Theorem 7.2), a result that is relevant to the study of representation of non-commutative C∗-algebras.
12 citations
TL;DR: In this article, the concept of Definable Factor Congruences (DFC) has been studied in the context of direct product representations of algebras in varieties, and several conditions expressing that these representations are definable in a first-order logic sense have been defined.
Abstract: We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.
12 citations
01 Jan 2012
TL;DR: In this paper, a linear vector equation is defined in terms of idempotent mathematics and an approach that is based on the analysis of distances between vectors in idemic vector spaces is applied to solve the equation.
Abstract: A linear vector equation is considered defined in terms of idempotent mathematics. To solve the equation, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces and reduces the solution of the equation to that of a tropical optimization problem. Based on the approach, existence and uniqueness conditions are established for the solution, and a general solution to the equation is given.
12 citations
TL;DR: The non-abelian idempotent theorem of as discussed by the authors is a quantitative version of the nonabelian non-abstraction theorem and can be taken to be triply tower in O(M) time.
Abstract: Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the non-abelian idempotent theorem.
12 citations
TL;DR: In this article, it was shown that there is a one to one correspondence between quantifiers and non-commutative binary operations making Q an idempotent and right-sided quantale.
Abstract: Let Q be an idempotent and right-sided quantale. There is a one to one correspondence between quantifiers and non-commutative binary operations making Q an idempotent and right-sided quantale. If Q is an atomic and irreducible orthomodular lattice there are only two such operations. Namely, the discrete quantifier and the indiscrete quantifer.
12 citations