Semigroups with Boolean congruence lattices

A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, respectively, with Boolean (...) congruence lattice are completely determined.     

Zero-sum problems with congruence conditions

For a finite abelian group G and a positive integer d, let s _dℕ(G) denote the smallest integer ℓ∈ℕ₀ such that every sequence S over G of length |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine s _dℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups G _p of G, the Davenport constant D(G _p ) is bounded above by 2exp  (G _p )−1. This generalizes former results for groups of rank two.     

Semigroups with complemented congruence lattices

Using the decomposition of a semigroup into itsI-classes, the paper gives a characterisation of all globally idempotent semigroups whose lattice of congruences is complemented. Furthermore, an arbitrary semigroup has a complemented congruence lattice if and only if it is an inflation of a semigroup characterised in this way. Thus the general problem of describing all semigroups with complemented congruence lattices is reduced to that of studying the question for the class of simple semigroups.Presented by Boris M. Schein.I wish to thank Professor Schein for his advice to study the work of TuUy ([15]) so that the original version of this paper could be shortened and simplified considerably.     

Orthodox semigroups with complemented congruence lattices

Communicated by J. M. Howie     

Positive algebras with nonetherian congruence lattices

Tashkent. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 2, pp. 181–185, March–April, 1992.     

Isotopic algebras with nonisomorphic congruence lattices

We present pairs of isotopic algebras with congruence lattices of different sizes, thus answering negatively the question of whether isotopic algebras must have isomorphic congruence lattices.     

Positive algebras with countable congruence lattices

Translated from Algebra i Logika, Vol. 31, No. 1, pp- 21–37, January–February, 1992.     

Scissor congruence

It is shown that certain simple figures can not be cut by scissors into pieces that can be reassembled to form certain other simple figures. The authors’ work was partially supported by National Science Foundation Grant G-14648 for Dubins, and Grant G-11594 for Hirsch.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

Finitary decidability implies congruence permutability for congruence modular varieties

We show that if a locally finite congruence modular varietyV is finitely decidable, thenV has to be congruence permutable.Presented by S. Burris.     

Properties of digraphs connected with some congruence relations

The paper extends the results given by M. Křížek and L. Somer, On a connection of number theory with graph theory, Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer n define a digraph Γ(n) whose set of vertices is the set H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a ³ ≡ b (mod n). The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph Γ(n) is proved. The formula for the number of fixed points of Γ(n) is established. Moreover, some connection of the length of cycles with the Carmichael λ-function is presented.      

Colourings with fixed congruence in finite projective planes

In this paper we study the colourings with fixed congruence (with respect to lines) in finite projective planes.