A special quartic congruence

D by a Bilinear Congruence of Twisted Elliptic Quartic Curves

On a Special Congruence of Carlitz

We prove that if q is a power of a prime p and p divides a, with k ≥ 0, then 1 + (q − 1) ∑ 0≤b(q−1)<a ( a b(q − 1) ) ≡ 0 (mod p). The special case of this congruence where q = p was proved by Carlitz in 1953 by means of rather deep properties of the Bernoulli numbers. A more direct approach produces our generalization and several related results.

Special Congruence Triples for a Regular Semigroup

With the usual notation for congruences on a regular semigroup S, in a previous communication we studied the lattice Λ generated by Γ = {σ, τ, μ, β} relative to properties such as distributivity and similar conditions. For K and T the kernel and trace relations on the congruence lattice of S, we form an abstraction of the triple (Λ;K|Λ, TΛ) called a c-triple. In this study a number of relations...

Simultaneous Congruence Representations: a Special Case

We study the problem of representing a pair of algebraic lattices, L1 and L0, as Con(A1) and Con(A0), respectively, with A1 an algebra and A0 a subalgebra of A1, and we provide such a representation in a special case. Back in 1971, Ervin Fried posed the problem of representing a pair of algebraic lattices, L1 and L0, as Con(A1) and Con(A0), respectively, with A1 an algebra and A0 a subalgebra o...

A Special Congruence Lattice of a Regular Semigroup

Let S be a regular semigroup and C its lattice of congruences. We consider the sublattice Λ of C generated by σ-the least group, τ -the greatest idempotent pure, μ-the greatest idempotent separating and β-the least band congruence on S. To this end, we study the following special cases: (1) any three of these congruences generate a distributive lattice, (2) Λ is distributive, (3) the restrictio...