  We find conditions on ideals of an algebra under which the algebra is
dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the
real numbers. We introduce a concept of bq-homomorphism (which is given by two
linear maps $f, g$ of the algebra to the set of the real numbers) and show that
an algebra is dibaric if and only if it admits a non-zero bq-homomorphism.
Using the pair $(f,g)$ we define conservative algebras and establish criteria
for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein
algebra and an algebra induced by a linear operator are introduced and
relations between these algebras are studied. For dibaric algebras we describe
a dibaric algebra homomorphism and study their properties by bq-homomorphisms
of the dibaric algebras. We apply the results to the (dibaric) evolution
algebra of a bisexual population. For this dibaric algebra we describe all
possible bq-homomorphisms and find conditions under which the algebra of a
bisexual population is induced by a linear operator. Moreover, some properties
of dibaric algebra homomorphisms of such algebras are studied.
