ON DOUBLING ALGEBRAS

Abstract

If (X; ∗) and (X; ◦) are binary systems then (X; ∗ )= ⇒ (X; ◦ )i f (x∗y)◦z = (x ∗ z) ∗ (y ∗ z) where (X; ◦) is the doubling algebra of the source algebra (X; ∗). Obviously there are many mutual influences on the types of (X; ∗) and (X; ◦). In this paper we investigate several of these mutual influences, including when (X; ∗ )i s a group, B-algebra, a cancellative semigroup with identity. Given a set X, let V (X) denote the collection of all binary algebras (or equivalently, groupoids) on X, i.e., V (X )= {(X; ∗) |∗ : binary operation on X}. An algebra (X; ∗ )i s said to be a source algebra of an algebra (X; ◦ )i f (x ∗ y) ◦ z =( x ∗ z) ∗ (y ∗ z), for any x, y, z ∈ X, and denoted by (X; ∗ )= ⇒ (X; ◦). In this case, we say (X; ◦) the doubling algebra of (X; ∗).