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.

Self-similarity and quasi-idempotence in neural networks and related dynamical systems.

Abstract: Self-similarity across length scales is pervasively observed in natural systems. Here, we investigate topological self-similarity in complex networks representing diverse forms of connectivity in the brain and some related dynamical systems, by considering the correlation between edges directly connecting any two nodes in a network and indirect connection between the same via all triangles spanning the rest of the network. We note that this aspect of self-similarity, which is distinct from hierarchically nested connectivity (coarse-grain similarity), is closely related to idempotence of the matrix representing the graph. We introduce two measures, ι(1) and ι(∞), which represent the element-wise correlation coefficients between the initial matrix and the ones obtained after squaring it once or infinitely many times, and term the matrices which yield large values of these parameters "quasi-idempotent". These measures delineate qualitatively different forms of "shallow" and "deep" quasi-idempotence, which are influenced by nodal strength heterogeneity. A high degree of quasi-idempotence was observed for partially synchronized mean-field Kuramoto oscillators with noise, electronic chaotic oscillators, and cultures of dissociated neurons, wherein the expression of quasi-idempotence correlated strongly with network maturity. Quasi-idempotence was also detected for macro-scale brain networks representing axonal connectivity, synchronization of slow activity fluctuations during idleness, and co-activation across experimental tasks, and preliminary data indicated that quasi-idempotence of structural connectivity may decrease with ageing. This initial study highlights that the form of network self-similarity indexed by quasi-idempotence is detectable in diverse dynamical systems, and draws attention to it as a possible basis for measures representing network "collectivity" and pattern formation.

Pseudo-loop conditions.

Abstract: About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form f ( x 1 , … , x n ) ≈ f ( y 1 , … , y n ) in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity s ( x , y , z , x ) ≈ s ( y , x , y , z ) in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form f ( x 1 , 1 , … , x 1 , n ) ≈ ⋯ ≈ f ( x m , 1 , … , x m , n ) , which we call loop conditions of width m . We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width m there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non-trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo-loop conditions of finite width, a generalisation suitable for non-idempotent algebras; they are of the form u 1 ∘ f ( x 1 , 1 , … , x 1 , n ) ≈ ⋯ ≈ u m ∘ f ( x m , 1 , … , x m , n ) , and of central importance for the structure of algebras associated with ω -categorical structures. We show that for the latter, satisfaction of a pseudo-loop condition is characterised by pseudo-loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo-loop condition exists (for ω -categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non-trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.

Idempotent Subreducts of Semimodules over Commutative Semirings

Abstract: A short proof of the characterization of idempotent subreducts of semimodules over commutative semirings is presented. It says that an idempotent algebra embeds into a semimodule over a commutative semiring, if and only if it belongs to the variety of Szendrei modes.