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- W2509392235 abstract "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 an- swer 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 con- sequence, 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 Implication zroupoids (I-zroupoids, for short) and show that I satisfies the identity x ''' → y ≈ x ' → y, where x ' := 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 x '' = x. The paper ends with some open problems for further research." @default.
- W2509392235 created "2016-09-16" @default.
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- W2509392235 date "2012-01-01" @default.
- W2509392235 modified "2023-09-27" @default.
- W2509392235 title "DE MORGAN ALGEBRAS: NEW PERSPECTIVES AND APPLICATIONS" @default.
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