  Conditional independence plays a foundational role in database theory,
probability theory, information theory, and graphical models. In databases,
conditional independence appears in database normalization and is known as the
(embedded) multivalued dependency. Many properties of conditional independence
are shared across various domains, and to some extent these commonalities can
be studied through a measure-theoretic approach. The present paper proposes an
alternative approach via semiring relations, defined by extending database
relations with tuple annotations from some commutative semiring. Integrating
various interpretations of conditional independence in this context, we
investigate how the choice of the underlying semiring impacts the corresponding
axiomatic and decomposition properties. We specifically identify positivity and
multiplicative cancellativity as the key semiring properties that enable
extending results from the relational context to the broader semiring
framework. Additionally, we explore the relationships between different
conditional independence notions through model theory, and consider how methods
to test logical consequence and validity generalize from database theory and
information theory to semiring relations.
