It is well known that Boolean algebras can be defined using
only the implication and the constant 0. It is, then, natural to ask
whether De Morgan algebras can also be characterized using only a binary
operation (implication) ¨ and a constant 0. In this paper, we
give an affirmative answer to this question by showing that the variety
of De Morgan algebras is term-equivalent to a variety of type {¨, 0}.
As a natural consequence, we describe Kleene algebras also as a variety
using only ¨ and 0. (The afore-mentioned result for Boolean algebras
is also deduced.) As a second consequence, we give a simplification of
an axiom system of Bernstein (along with a new proof for his system
of axioms). We also describe De Morgan algebras in terms of a NAND
operation | and the constant 0. Motivated by the the afore-mentioned
results, we define, and initiate, the investigation of a new variety I of algebras,
called gImplication zroupoidsh (I-zroupoids, for short) and show
that I satisfies the identity x0000 ? x00, where x0 := x ¨ 0. Furthermore,
we introduce several important subvarieties of I and establish some relationships
among them; in particular, we give several characterizations
of the subvariety defined by x00 = x. The paper ends with some open
problems for further research.