We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. consider the monoid generated by such functions. call idempotent element this monoid. They are interval retracts. Some them realize kind parabolic map called projections. prove that, in Eulerian posets, image projection, its complement, induced subposets. In Coxeter group, all ...

The element $q$ of a ring $R$ is called quasi-idempotent if $q^2=uq$ for some central unit $u$ $R$, or equivalently $q=ue$, where and $e$ an idempotent $R$. In this paper, we define that the almost quasi-clean each sum regular element. Several properties almost-quasi clean rings are investigated. We prove every quasi-continuous nonsingular quasi-clean. Finally, it determined conditions under wh...

If we consider the real numbers extended by minus infinity with the operations maximum and addition, we obtain the max-algebra or the max-plus semiring. The analog of linear algebra for these operations extended to matrices and vectors has been widely studied. We outline some facts on semirings and max-plus linear algebra, in particular, the solution of maxplus linear systems. As an application...

A Q-system in a C? 2-category is unitary version of separable Frobenius algebra object and can be viewed as higher idempotent. We define idempotent completion for 2-categories called study its properties. show that the right correspondences unital C?-algebras complete by constructing an inverse realization †-2-functor. use this result to construct induced actions group theoretical fusion catego...