This article develops and tests an algorithm of spatial congruence based on geometric congruity of two spatial areal objects in the Euclidean plane. Spatial congruence is defined and thus evaluated as an increasing continuous function of congruity in the position, orientation, size, and shape of spatial objects, dependent upon scaling, translation, and rotation. Expansion-based geometric matchi...

A systematic approach is developed for enumerating congruence classes of 2-cell imbeddings of connected graphs on closed orientable 2-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration.

If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V . We give some criteria for the first-order axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence n-permutable variety satisfies a nontrivial congruence identity.

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.