We establish a congruence modulo four in the real Schubert calculus on the Grassmannian of m-planes in 2m-space. This congruence holds for fibers of the Wronski map and a generalization to what we call symmetric Schubert problems. This strengthens the usual congruence modulo two for numbers of real solutions to geometric problems. It also gives examples of geometric problems given by fibers of ...

We define an 2-fuzzy congruence, which is a weakened fuzzy congruence, find the 2-fuzzy congruence generated by the union of two 2-fuzzy congruences on a semigroup, and characterize the 2-fuzzy congruences generated by fuzzy relations on semigroups. We also show that the collection of all 2-fuzzy congruences on a semigroup is a complete lattice and that the collection of 2-fuzzy congruences und...

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving ex...

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347–353], based on Sergeichuk’s paper [Math. USSR, Izvestiya 31 (3) (1988) 481–501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Ou...