A canonical form for congruence of matrices was introduced by Turnbull and Aitken in 1932. More than 70 years later, in 2006, another canonical form for congruence has been introduced by Horn and Sergeichuk. The main purpose of this paper is to compare both canonical forms and provide a brief survey on the history of the canonical form for congruence.

In this paper congruences on orthomodular lattices are studied with particular regard to analogies in Boolean algebras. For this reason the lattice of p-ideals (corresponding to the congruence lattice) and the interplay between congruence classes is investigated. From the results adduced there, congruence regularity, uniformity and permutability for orthomodular lattices can be derived easily.

We describe an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a nontrivial congruence identity. Our results yield slight strengthenings of Day’s Theorem and Gumm’s Theorem, which each characterize congruence modularity.

We previously obtained a congruence modulo four for the number of real solutions to many Schubert problems on a square Grassmannian given by osculating flags. Here, we consider Schubert problems given by more general isotropic flags, and prove this congruence modulo four for the largest class of Schubert problems that could be expected to exhibit this congruence.

We summarize the combinatorial properties of congruence generation in congruence distributive varieties which are relevant to Baker’s finite basis theorem, explain the extent to which these properties survive in congruence meet-semidistributive varieties, indicate our approach to extending Baker’s theorem to the latter varieties, and pose several problems which our approach does not answer.

We develop a version of stochastic Pi-calculus with a semantics based on measure theory. We define the behaviour of a process in a rate environment using measures over the measurable space of processes induced by structural congruence. We extend the stochastic bisimulation to include the concept of rate environment and prove that this equivalence is a congruence which extends the structural con...

We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson’s congruence criterion for torsion free λ-rings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups and Frobenius isogenies.

We call a lattice L isoform, if for any congruence relation Θ of L, all congruence classes of Θ are isomorphic sublattices. We prove that for every finite distributive lattice D, there exists a finite isoform lattice L such that the congruence lattice of L is isomorphic to D.

Contextual equivalence equate terms that have the same observable behaviour in any context. A standard contextual equivalence for CCS is the strong barbed congruence. Configuration structures are a denotational semantics for processes in which one define equivalences that are more discriminating, i.e. that distinguish the denotation of terms equated by barbed congruence. Hereditary history pres...

For an anti-congruence q we say that it is regular anti-congruence on semigroup (S,=, =, ·, α) ordered under anti-order α if there exists an antiorder θ on S/q such that the natural epimorphism is a reverse isotone homomorphism of semigroups. Anti-congruence q is regular if there exists a quasi-antiorder σ on S under α such that q = σ ∪ σ−1. Besides, for regular anti-congruence q on S, a constr...