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...

Let T be the set of ground terms over a nite ranked alphabet We de ne partial automata on T and prove that the nitely gener ated congruences on T are in one to one correspondence up to iso morphism with the nite partial automata on T with no inaccessible and no inessential states We give an application in term rewriting every ground term rewrite system has a canonical equivalent system that can...

Let A be a recursive structure, and let ψ be a recursive infinitary Π2 sentence involving a new relation symbol. The main result of the paper gives syntactical conditions which are necessary and sufficient for every recursive copy of A to have a recursive expansion to a model of ψ, provided A satisfies certain decidability conditions. The decidability conditions involve a notion of rank. The ma...

For a givenweighted finite automaton over a strong bimonoid we construct its reducedNerode automaton, which is crisp-deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics.We show that the reducedNerode automaton is even smaller than theNerode automaton, which was previously used in determinization related to this semantics. We determine ne...

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.