A class F of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ F is in F ; 2) If α1, α2 are congruences on A and A/αi ∈ F , i = 1, 2, then A/(α1∩ α2) ∈ F . We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruen...

We introduce a novel cascade demixing structure for multichannel blind deconvolution in nonminimum phase systems. To simplify the learning process, we decompose the demixing model into a causal finite impulse response (FIR) filter and an anticausal scalar FIR filter. A permutable cascade structure is constructed by two subfilters. After discussing geometrical structure of FIR filter manifold, w...

In this note we show that if two critical algebras generate the same congruence-permutable variety, then the varieties generated by their proper sections also coincide.

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type also satisfy $f\circ g =g \circ f$. We show if the fast escaping sets those functions contained their respective Julia then must have same set. obtain conclusion about commuting quasimeromorphic with infinite backward orbit infinity. Furthermore we permutable form $g=\phi\circ f$, where $\phi$ is a quasic...

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras h...