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.

Nil-clean quadratic elements

Abstract: We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

On certain classes of rings of formal matrices

Abstract: We consider a problem on isomorphism of rings of formal matrices of order three with values in a ring R. We study conditions of regularity and complete idempotence for rings of generalized matrices.

Combining Equational Tree Automata over AC and ACI Theories

Abstract: In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ imply similar decidability results in the combined theory${\mathcal{E}}_1 \cup {\mathcal{E}}_2$; (2) checking emptiness of a language obtained from the Boolean combinationof regular equational tree languages. We present a negative result for the first problem. Specifically, we show that the intersection-emptiness problem for tree automata over a theory containing at least one AC symbol, one ACI symbol, and 4 constants is undecidable despite being decidable if either the AC or ACI symbol is removed. Our result shows that decidability of intersection-emptiness is a non-modularproperty even for the union of disjoint theories. Our second contribution is to show a decidability result which implies the decidability of two open problems: (1) If idempotence is treated as a rule f(x,x) ?xrather than an equation f(x,x) = x, is it decidable whether an AC tree automata accepts an idempotent normal form? (2) If ${\mathcal{E}}$ contains a single ACI symbol and arbitrary free symbols, is emptiness decidable for a Boolean combination of regular ${\mathcal{E}}$-tree languages?

Assessment of Smart Grid Systems for Electricity Using Power Maclaurin Symmetric Mean Operators Based on T-Spherical Fuzzy Information

Abstract: Traditional electricity networks are replaced by smart grids to increase efficiency at a low cost. Several energy projects in Pakistan have been developed, while others are currently in the planning stages. To assess the performance of the smart grids in Pakistan, this article employs a multi-attribute group decision-making (MAGDM) strategy based on power Maclaurin symmetric mean (PMSM) operators. We proposed a T-spherical fuzzy (TSF) power MSM (TSFPMSM), and a weighted TSFPMSM (WTSFPMSM) operator. The proposed work aims to analyze the problem involving smart grids in an uncertain environment by covering four aspects of uncertain information. The idempotency, boundedness, and monotonicity features of the proposed TSFPMSM are investigated. In order to assess Pakistan’s smart grid networks based on the suggested TSFPMSM operators, a MAGDM algorithm has been developed. The sensitivity analysis of the proposed numerical example is analyzed based on observing the reaction of the variation of the sensitive parameters, followed by a comprehensive comparative study. The comparison results show the superiority of the proposed approach.