Dellalgebra

Dellalgebra is a term used in abstract algebra to denote a class of algebraic structures that generalize systems with two interacting operations. While definitions vary, most treatments describe a dellalgebra as a set A equipped with two binary operations: an addition and a second operation called dell, often written with a dot or by juxtaposition. The precise axioms are chosen to suit different mathematical contexts.

A typical dellalgebra imposes that (A, +) is an abelian monoid with identity 0, and that (A, ·) is associative. The two operations interact via a distributive law: a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c for all a, b, c in A. Some authors allow nonassociative dell or relax one or both distributive laws, yielding a family of variants used in various mathematical settings.

Examples commonly cited include the natural numbers with ordinary addition and multiplication, which satisfy the standard semiring axioms and thus form a dellalgebra under this framework. More generally, any semiring or tropical semiring can be viewed as a dellalgebra when interpreted through the two-operation lens. In theoretical computer science, dellalgebras appear in automata theory and formal language study as abstract models of computation and resource combination.

History and usage: The term appears primarily in niche mathematical literature and as a pedagogical device illustrating how changing axioms affects structure. Dellalgebras relate to semirings, rings, and algebras, and are discussed alongside tropical algebra and category-theoretic approaches to duality.