Description Of Congruences Research Articles

Perfect congruences on bisimple $$\omega $$ ω -semigroups

A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $\omega$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $\omega$-semigroups, substantially generalizing the above mentioned result of \cite{key7}.

Congruences of Dislocations in Continuously Dislocated Crystals

The time-dependent congruences of Volterra-type dislocations are investigated based on the generalized formulas of Frenet in a Riemannian space. The analysis is applied to the description of congruences of edge and mixed dislocations consistent with a continuous distribution of dislocations for which its material space is an equidistant Riemannian space. In particular, the principal congruences of dislocations are considered. The kinematics of congruences of mixed dislocations endowed with univocally defined local slip planes is discussed. It is shown that the geometry of such congruences of dislocations admits a class of nonlinear evolution equations describing the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions imposed on the derived system of equations in order to describe the evolution of curvature and torsion of congruences of edge dislocations are proposed. In the static case, an expression is given for shear stresses required to bend prismatic edge dislocations of torsion zero located on the totally geodesic crystal surfaces. It follows that the congruence of these dislocations is endowed with a finite self-energy function.

The congruence lattice of a Tully semigroup

This paper presents a description of congruences on Tully semigroups by means of congruence aggregates. Formulas for join and meet of arbitrary sets of congruences are developed. As an application, necessary and sufficient conditions on the parameters of a Tully semigroup in order that its congruence lattice be contained in a given variety of modular lattices are obtained.